MATHEMATICAL TRACTS. eCambtbg e: PRINTED BY C. J. CLAY, M.A. AND SONS, AT THE UNIVERSITY PRESS. MATHEMATICAL TRACTS. PART I. BY F. W. NEWMAN, M.R.A.S. EMERITUS PROFESSOR OF UNIVERSITY COLLEGE, LONDON; HONORARY FELLOW OF WORCESTER COLLEGE, OXFORD. e~ambrige: MACMILLAN AND BOWES. 1888 [Ail Rights reserved.] TRACT I. ON THE BASES OF GEOMETRY WITH THE GEOMETRICAL TREATMENT OF J-1. CONTENTS. (1) On the Treatment of Ratio between Quantities Incommensurable. (2) Primary Ideas of the Sphere and Circle. Poles of a Sphere. (3) Definition and Properties of the Straight Line. (4) Definition and Properties of the Plane. (5) Parallel Straight Lines based on the Infinite Area of a Plane Angle. (6) On the Volume of the Pyramid and Cone. I. THE RATIO OF INCOMMENSURABLES. 1. IN arithmetic the first ideas of ratio and proportion, and the laws of passage from one set of 4 proportionals to another, ought to be learned, as preliminary to geometry; but in geometry the doctrine of incommensurables requires a special treatment, unless the learner be well grounded in the argument of infinite converging series. Repeating decimals may perhaps suffice. Another, possibly better way, is open.by the introduction of VARIABLE quantities, which will here be proposed. 2. Nothing is simpler than to imagine some geometrical quantity to vary in shape or size according to some prescribed law. This must imply at least two quantities varying together. Thus, if an equilateral triangle change the length of its side, its area also changes. If the radius of a circle increase or diminish, so does the length of the circumference. In general two magnitudes X and Y may vary together: they may be either the same in kind,-as the radius and circumference of a circle is each a length; or the two may be different in kind, say, a length and an area. In general it is a N. 1 2 VARIABLES WHICH INCREASE UNIFORMLY. convenient notation to suppose that when X changes to X', Y changes to Y'. 3. Again, if X receive successive additions x,,x,...x *, the corresponding additions (if additions they be) to Y are well denoted by yyy3... y. An obvious and simple case, if it occur, will deserve notice; namely, if the two variables are so regulated, that equality in the first set of additions (i.e. x,= =,=... = x) induces equality in the second set; (i.e. y,= y,= ys=.. = y,). The variables X and Y are then said to increase uniformly. As an obvious illustration, suppose X to be the arc of a circle, and Y the area, x of the sector which it bounds, evidently then if ~ the arcs x,, x, are equal increments of the arc X, Y the sectors y1y, which are bounded by xlx, will be equal increments of Y. Then the arc X and the sector Y increase together uniformly. o 4. We may now establish a theorem highly convenient for application in geometry, alike whether quantities are commensurable or incommensurable. THEOREM. "If X and Y are any two connected variables, which begin from zero together, and increase uniformly; then X varies proportionably to Y. In other words, if Y become Y' when X becomes X', then X is to X' as Y is to Y'." Proof. First, suppose X and X' commensurable, and e a common measure, or X= m.: (m times e) and X' = n:. We may then suppose X and X' made up by repeated additions of I. Every time that X has the increment e, Y will receive a uniform increment which we may call v; then Y is always the same multiple of v that X is of e; thus the equation X = mn implies Y= mv, and X' =hn: implies Y'= nv. Hence X: X'= m: n = Y: Y'. Next, when X' is not commensurate with X, yet e is some submultiple of X, such that n = X, and X' contains e more than mz times, but less than (m + 1) times; evidently we cannot have X: X'=Y: Y' (when the four magnitudes are presented to us) unless, as a first condition, on assuming nv= Y, we find Y' to contain v more than m times and less than (m + 1) times: and unless this condition were fulfilled, X and Y would not increase uniformly. We may therefore DISTANCE OR SHORTEST PATH. 3 assume X2X3 on opposite sides of X', with values X = m, X3 = (m + 1); likewise Y2Y3 on opposite sides of Y', with values Y2 = mv, Y3= (mnz + 1) v. Then by the first case we have X: X2 = Y: Y2 and X: X3= Y: Y3. But X3-X2=:, and Y3-Y2=v. Let n perpetually increase, then % and v perpetually lessen. X2 and X3 run together in X', Y and Y3 run together in Y'. Thus each of the ratios X: X2 and X: X3 falls into X: X', and each of the ratios Y: Y2, Y: Y3 falls into Y: Y'. Inevitably then, X: X'= Y: Y', even when these last are incommensurate. Q.E.D. II. PRIMARY IDEAS OF THE SPHERE AND CIRCLE. For the convenience of beginners, POSTULATES may be advanced concerning the straight line and the plane, as well as concerning parallel straight lines. But in the second stage of study the whole topic ought to be treated anew from the beginning: a task which is here assumed. On Length and Distance. THEOREM. "All lengths are numerically comparable." To make this clear, it is simplest to imagine a thread indefinitely thin, flexible and inextensible. This, if applied upon any given line, will become an exact measure of its length; and if any two lines be then measured by two threads, the threads are directly comparable, shewing either that they are equal, or that one is longer than the other and how much longer. Hereby we safely assert the same fact concerning any two given lengths. Obviously, length is continuous magnitude: which means, that if a point P run along from A to B, the length AP passes through all magnitude from zero to AB. THEOREM. Any two given points in space may be joined either by one path which is shorter than any other possible, or by several equal paths than which none other is so short. For of all possible paths joining them some must be needlessly long; yet unless there is some limit to the shortening, the distance would be nil; the points would not be two, but would coincide and become one. 1-2 4 THE SPHERE. DEF. A shortest path that joins two points in space gives a measure of their DISTANCE. The same argument applies, if the two given points and the line that joins them must lie on a given surface; or again, if two surfaces that do not touch be given, and we speak of the shortest distance of the two surfaces. Assume a fixed point A and a second point S so movable as always to be at the same distance from it. It will be able to play all round A: therefore its locus will be a surface enclosing A. The solid mass enclosed is called a SPHERE (Globe or Ball) and A its CENTRE. THEOREM. "Every point outside the sphere is further from the centre and every point within the sphere is nearer to the centre, than are the points on the surface." For if T be an exterior point, every path joining T to A must pierce the surface in some point S; therefore the path TSA is longer than SA by the interval TS. Again, if R be within the sphere, we may imagine an interior sphere whose surface is at the common distance AR from A. Then S being exterior to the new sphere, SA is longer than RA; that is, R within the sphere of S is nearer to A than is the locus of S. Q. E. D. DEF. Two such concentric spheres enclose within their surfaces a solid called a spherical shell. THEOREM. "The two surfaces are equidistant, each from the other." For if the shortest distance from a point S to the inner surface is the path SR, symmetry all round shews at once that if from a second point S' the shortest path will be S'R', the two distances SR, S'R' will be equal. Indeed it is not amiss to remark, that if any spherical surface be rigidly attached to its centre, the entire surface may glide on its own ground without disturbing its centre, because the distances SA, S'A nowhere change. Hence also we may justly imagine the spherical shell to glide on its own ground, while the centre suffers no displacement, and any shortest path S'R' joining the opposite sides of the shell may assume the place which was previously held by SR. Actual superposition thus attests equality of distance. THEOREM. "If a spherical surface be given, its centre is determined." For if an inner point R be assumed at a given distance D from the surface, its locus is an interior continuous surface. Within this, at distance D', imagine a point R' to generate a second con POLES AND PARALLEL CIRCLES. 5 tinuous surface, and it will bé interior to the preceding and so continually. The series of surfaces must then necessarily converge towards a single point, which will be the centre of the given surface, because the sum of the distances is the same, from whichever point we calculate. The same argument proves that all the surfaces are concentric spheres. Poles of a Sphere. THEOREM. "To every point on a sphere one opposite point lies at the longest distance along the surface." For if the point P be given, and we take a point S at any distance from P along the surface, and suppose S to vary under the sole condition that its distance from P (along the surface) shall not change, the locus of S is a selfrejoining line enclosing P. (We call this a circle.) Next, beyond S, along the surface, take a new point T, which moves without changing its distance from S and froin P. This generates an outer circle, cutting off a part of the surface which was beyond the circle of S. Beyond this we may similarly form a third circle, and this series of circles ever lessening the finite area beyond it, will necessarily converge towards a point Q on the sphere. P will then be farther from Q (along the sphere) than any of these parallel circles. We call P and Q opposite poles of the sphere. The distance between them is evidently the half girth of the sphere. Every point on the sphere has not only its own opposite pole; but also its system of equidistant (or parallel) circles. The middle one of these (that is, the one equidistant from the two poles), is called their equator. If in an equator whose poles are P and Q, you fix any point C, and then proceeding half round the equator fix a second point D, C and D are evidently opposite poles. If you imagine a sphere to glide on its own ground, with centre unmoved, you may suppose P to pass over to the site held previously by Q. This carries Q to the place previously held by P. Thus the poles are exchangeable, while the sphere as a whole is unchanged and the same equator is attained. THEOREM. "If P and R be any two points on a sphere that are not opposite poles, one equator, and one only, passes through them both." TiHE STRAIGHT LINE. Proof. Through P and its opposite pole Q (just as above through the poles C and D) an equator may pass. If this half equator PQ become rigid and be rigidly attached to the fixed centre A, it still may sweep over the spherical surface (without change of P or Q) until it passes through R; but after passing once through R, it does not come back to it, except in a second revolution. Q. E. D. III. POINTS LYING EVENLY. In Simson's Euclid, the line whose points lie evenly is called STRAIGHT; but the phrase "lying evenly" is not explained. We can now explain it. When the two poles P and Q, and the centre A, all remain unchanged, nevertheless each of the parallel circles associated with P and Q can glide on their own ground. Evidently then, if P and A be fixed, this suffices to fix Q. In fact while each circle spins round its own line, Q can only spin round itself. Also, to fix P and Q fixes A.-These parallel circles excellently define to us the idea of rotation, which is a constrained motion, still possible, even when P, A, Q are all fixed. Now suppose that a line PMQ internal to the sphere rigidly connects P with Q. Then if the system revolve round P,, Q, PMQ may generate a self-rejoining surface within the sphere. Again within this new surface a rigid line PNQ may connect P with Q, and the line PNQ by rotation round P, A, Q may generate a third surface interior to the preceding; and so on continually. Since there is no limit to the constant thinning of the innermost solid, we see that a mere line without thickness connects P with Q and passes through A, which line is interior to all the solids and during rotation remains immovable. It is called an axis, and can only turn about itself. Hence every point in this axis lies evenly between P and Q. And since P and Q may represent any two points in space, we now discover that between any two there is a unique line lying evenly. This continuous line, while we talk of rotation round it, is entitled an axis; but ordinarily we call it simply STRAIGHT. On the Straight Line and its " Direction." We now infer that 1. Any two points in space can be joined by a straight line, 2. Every part of a straight line is straight. THE PLANE. 7 3. A unique straight line is determined, when its two end-points are given. 4. Any part of a straight line, if removed, may take the place of any equal part of the same. Hence it easily follows that a straight line, gliding along itself, will prolong itself indefinitely far, either way, along a determinate course. We are now able to sharpen our idea of direction. Hitherto we might say vaguely, "Imagine a path to proceed in any direction," that is, without particular guidance. But now we see, that if ever so short a straight line be drawn, it points to a definite prolongation beyond itself, of indefinite extent. This we entitle its direction. If this direction be changed, a deviation there occurs, and a sharp corner is recognized at the point of deviation. The amount of deviation suggests a new kind of magnitude, which will presently need attention. Now it suffices to remark on the case in which a new line AZ deviates equally from a previous line PA and from AQ the prolongation of PA. The equality is tested by imagining AZ to become an axis of a sphere. Then if P and Q revolve in the same circle, ZA is equally inclined to AP and to AQ. It is called perpendicular to PAQ. Evidently Z (on the sphere) is at the distance of a quarter girth from every point of the equator traced by P and Q. IV. THE PLANE. We return to the sphere. When any two poles P, Q are joined by a straight line, it has been seen that this passes through the centre A. The line PAQ is called a diameter of the sphere, and its half (AP or AQ) is called a radius. Evidently all the radii of the same sphere are equal; and of different spheres the greater the radius, the greater the sphere. If an equator CDEC is midway between the poles P, Q, and D is the pole opposite to C, then as the diameter PQ, so too the diameter CD, passes through centre A. This is true, whatever point in the equator is assumed for C. Therefore CAD is a varying diameter, whose extremities trace out the equator, while the diameter traces out a surface in which the equator lies. This surface is called a PLANE, and in particular is the plane of the equatorial circle. It was seen that P and Q might exchange places, while the centre A, and the sphere's surface as a whole, remain unchanged. Necessarily also the plane of the equator remains unchanged. It is 8 THE PLANE RULED BY A STRAIGHT LINE. then symmetrical on its opposite sides, or in popular language, the plane turns the sameface towards P as towards Q. The axis PAQ is called perpendicular to the plane of the equator, being perpendicular to every radius of the equatorial circle. THEOREM. " No other line but AP can be perpendicular to the plane of the equator." Proof. For if AR be some other radius of the sphere, some one of the parallel circles, whose pole is P, passes through R, and every point of this circle is nearer to the equatorial circle than is the pole P. Therefore the distance of R from the equator is less than a quarter of the sphere's girth, a fact which shews RA not to be perpendicular. THEOREM. " Through any two radii AP, AR of a sphere, that are not in the same straight line, one plane and one only may pass." It has been seen that through P and R only one equator can pass. The plane of this equator is the plane that passes through the two radii. Cardinal Property of the Plane. THEOREM. "If M and N are any two points in a plane, no point in the straight line which joins M and N can lie off the plane on either side." Symmetry suffices to establish this truth. Our hypothesis supplies data to fix what line is meant by MN, but gives no reason why any point of it should lie off the plane on one side rather than on the other; for the whole line is determined by merely the extreme points M, N, of which neither can guide any point towards P rather than towards Q. Thus there is no adequate reason, for deviation towards either side. Symmetry of data is in other mathematical topics accepted as an adequate argument for symmetry of results. Otherwise, "the want of sufficient reason for diversity" passes as refutation of alleged diversity. Therefore the argument here presented has nothing really novel. We have now a new method of generating a plane that shall pass through two intersecting straight lines LM, MN. Along ML let a point E run, and along MN similarly a point F. Join EF while the motion of E and of F continues. Then EF (by the last THE PLANE EVERYWHERE LIKE ITSELF. 9 Theorem) always continues to- rest on the plane LMN. This mode of generating the plane supersedes the idea of rotation. For simplicity we might suppose ME: MF to retain a fixed ratio. THEOREM. " A plane has no unique point or centre." For if we start from given spherical radii AP, AR through which passes an (equatorial) plane, in AP take M arbitrarily, and in AR take N arbitrarily. Then we have seen that the locus of the moving line MN is our given plane. But again, in this plane take a fixed point 0, and join O to fixed points M and N. Then from the lines OM, ON we can (as in the last) generate the very same plane, which can glide on its own ground as the sphere did; thus the point A can pass to O without changing the ground or surface as a whole. The plane is infinite, the sphere is finite; but as with the sphere, so with the plane, no point of the surface is unique. After this, no impediment from logic forbids our passing to the received routine of Plane Geometry, until we are arrested by the difficulty of parallel straight lines, to which I proceed, after one remark on the definition of an angle. Above, a sharp corner or turn was identified with deviation, or change of direction. In geometry it has the name of an angle, and we measure its magnitude by aid of the circular arc which it subtends at the centre or by the sector of that arc. But no insuperable logic forbids our estimating the magnitude of an angle by the portion of the infinite area which it intercepts from a plane; which indeed is suggested by a perpetual elongation of the radius of the circle whose sector was assumed as measure of the angle. Monsieur Vincent in Paris (1837) adopted this definition as adequate to demonstrate the equivalent of Euclid's Twelfth Axiom without any new axiom at all. Has this method received due attention in England? Monsieur Vincent was not the first to suggest accepting the infinite plane area cut off by two intersecting straight lines, as the measure of the angle which they enclose: but perhaps he was the first to introduce the method into a treatise on Elementary Geometry, that obtained acceptance in so high an institution as the University of France. Two lemmas alone are wanted, and these every beginner will find natural. 10 PROOF OF THE TWELFTH AXIOM. LEMMA I. "Every angle is a finite fraction of a right angle;" that is, some finite multiple of it exceeds 90~. For the circular arc which subtends it, is always some finite fraction of the quarter of the circumference. DEF. When two straight lines AM, BN in the same plane are both perpendicular to a third straight line AB, we call that portion of the plane area which is enclosed between MA, AB and BN a BAND. LEMMA II. Then, I say, whatever the breadth (AB) of the band, the area of the band is less than any finite fraction of a right angle. Proof. Prolong AB indefinitely to X, and along it take any number of equal lengths AB = BC = CD = DE, &c., and through C, D, E... draw perpendicular to ABCDE... straight lines CO, DP, EQ, &c. Evidently then the successive bands are equal, by superposition. Thus, whatever multiple of the first band be deducted frorn the plane area marked off by the right angle MAX, the loss is insensible; for, as remainder, we find the area marked off still by a right angle (such as QEX, if only four bands were deducted). Any two right angles embrace areas which can be identified by superposition, and have no appreciable difference. The matter may be concisely summed up by remarking, that every band is infinite in one direction only,-say, horizontally-but the area embraced by any right angle is infinite in both directions, horizontally and also vertically. Thus it is no paradox to say, that no finite multiple of the band can, by its deduction from the area of the right angle, lessen that infinite area in our estimate. Q.E.D. Euclid's Twelfth Axiom is now an immediate corollary; viz. If MABN be any band; and, within the right B angle NBA, any straight line BT be drawn, T' it can be prolonged so far as to meet the prolongation of AM. For the angle NBT is a finite fraction of a right angle, while the band MBAN is less than any finite fraction of the same; hence the angle NBT is greater than the band MABN, but unless BT crossed AM this would be false. Thus of necessity the two lines do cross, as we asserted. I cannot see any new axiom involved in this proof: therefore I am forced to abandon several other specious methods and give it VOLUME OF PARALLELEPIPEDON. 11 preference. Surely we may bow to the authority of the University of France in such a matter. On the Volume of Pyramids and Cones. The treatment of this topic in Euclid is very clumsy. It demands and it admits much improvement. 1. For parallelepipeda prove first, that if two such solids differ solely in the length of one edge, which we may call x in the one and a in the other, then their volumes are in the proportion of X: a. 2. Next, if they have a solid angle in common, but the edges round it are in one x, y, z, and in the other a, b, c, then the two volumes are in the proportion of xyz: abc. 3. After this it is easily shewn that parallelepipeda on the same base and equal height have equal volumes. 4. Therefore finally, that the volume of a parallelepipedon is measured by its base x its height. COR. The same is true of any prism. From this we proceed to approximate to the volume of a pyramid. 5. Divide the height (h) into (n) equal parts by (n - 1) planes all parallel to the base (B). Establish, on these (n- 1) bases, upright walls, and you will find you have constituted a double system of prisms, one interior to the pyramid, one exterior; the latter has the lowest prism in excess of the other system. Every base is similar to every other, by the nature of a pyramid. The volume here of every prism is - x its base, the number n and h being the same for all, but the base varying. 6. The base whose distance from the vertex is r. h, is to the n r2 original (B) as r2 n2; hence its area is. B, which gives for the volume of the prism standing on it (2... Hence the sum of th volumes of th eternal prisms i12+22+32 +...+ n2 h d the volumes of the external prisms is B, and by omitting n2 from the numerator of the larger fraction we obtain 12 VOLUME OF PYRAMID. n2.h.B the sum of volumes for the internal prisms. Now since vanishes when h, B are finite and n infinite, the difference of the two systems of prisms vanishes when n is infinite. But the volume of the pyramid is less than the exterior system and greater than the interior; hence each system has the volume of the pyramid for its limit, when n increases indefinitely. 7. Let / be the unknown numerical limit to which the fraction 12 + 2 + 32 +...e2 1 3 + —~ approximates when n thus increases. Then the volume of the pyramid =. h. B. Since ~ is the limit of a numerical fraction, which remains the same, whatever the form of the pyramid's base, we shall know the value of, for all pyramids, if we can find it in one. Meanwhile the result V=Y.h.B at once shews that pyramids with equal base and equal height have equal volume, since, is the same for all. 8. When this theorem has been attained, we have only to divide a triangular prism into three pyramids, and instantly infer that the 3 are equal among themselves; therefore that each has a volume just ^ of the prism, i.e. equal lh. B. This, being proved of a pyramid whose base is a triangle, shews that the unknown.~ is there exactly 1. Hence universally, = -, and volume of every pyramid = lh. B, or is equal to ~ of the prism which has the same base and height. COR. Every cone also is one third of the cylinder which has the same base and height. TRACT II. GEOMETRICAL TREATMENT OF /-1. 1. In pure algebra, concerned with number only, the symbols + and -, denoting addition and subtraction, in an early stage needed elucidation when the mark of minus was doubled. It is found natural that -(+ a) and +(- a) should both mean - a, but that -(-a) should mean + a, and (-a). (- b) should be + (ab) surprises a beginner, and is illustrated by urging that to subtract a debt increases the debtor's property, and to subtract cold is to add heat. But as soon as we apply algebra to geometry, the symbols + and - are still better interpreted of reverse direction; also time past and time coming afford equally good illustration. Distinguishing positive and negative direction along a line, we find no mystery in the fact that to reverse negative direction is to make it positive, so that - (- a) gives + a, as reasonably as - (+ a) gives - a. If we know beforehand whether a given distance is to be counted positively or negatively along a given axis, no ambiguity is incurred, and the sign + or - generally gives the needfill information. For this reason some are apt to think of +a and -a as different numbers, instead of the same number differently directed. Out of this rises the learner's natural complaint when he meets \/- 1 or 4/- 5. " There is no such number: you confess it is imaginary: a proposition involving it has no sense." So murmurs every scrupulous and wary beginner: and the teacher's reply, "Somehow we work out useful results by /- 1," sounds like saying: "Out of this nonsense useful truth is elicited." 2. The first reply to be made is: No one ought to desire any number for A/-1 except the unit itself; the &/- which precedes, though a double symbol, has the force of a symbol only. The next reply is decisive,-the double symbol \/- points to a new direction in geometry; namely, the direction perpendicular to + 1 and -1. But to explain this fully, it is better to make a new beginning. 14 WARREN'S RADII, Suppose that radii issue in many directions from a fixed point in a plane, and that distances are counted along them. So long as we know along which radius we are to count, nothing new is involved, and of course no difficulty. Suppose one of these radii to be our ordinary positive axis, another to be called the m radius, and for distinction write the index m under every number to be counted along it, so that 1, is its unit, in length = 1 as estimated absolutely. Then we deal with amb,,cm... along the m line, and combine a,, + b,, and interpret 5a,,,, 5b,, ~ 6c,, without difficulty or fear, since all are lengths to be counted along the same radius. But such a product as a,. b, would need careful interpretation. In ab. no obscurity is found, whether the a be linear or numerical. If linear, we proceed as in interpreting ab3, though space has only three dimensions. If we put A for the value of ab3, we attain it by the proportion 1: a = b: A, so that A is the same in kind as b3. Similarly if A'=ab,,, we are able to count A' along the n radius, whether a be simply numerical, or when it is linear, by aid of the ratios 1:a = b: A'. But if we proceed thus with an. b., using the proportion 1m: a=b,: A', we confound amb, with ab.; for 1m: a is the same ratio as 1: a. 3. Mr Warren in 1826 laid a logical basis for this matter by his treatise on /- 1, which I here substantially follow, and wonder that it is not found in all elementary works. He virtually distinguishes between proportionate lengths and proportionate lines. In the former, DIRECTION is not regarded; with the latter, it is essential. Thus if A, B, C, D are proportionate lengths, but are drawn along our radii, -viz. A along the positive axis, B along the m radius, C on the n radius and D on the p radius, we do not pronounce these lnes proportional, unless also their directions justify it; that is, the p radius must be disposed towards the n radius, as is the m radius to the positive axis. This amounts to saying that the p line must lie on the same side of the n line, and at the same angular distance from it, as the m line com- p N pared with the positive axis. Then, if OL, M OM, ON, OP be the 4 radii, and LMNP a cir- R cular arc, we need that the arc PN shall =arc ML before we admit that the units o L OL, OM, ON, OP are proportionate lines. After this condition of the directions is fulfilled, we concede that WHEN PROPORTIONAL. 15 the proportionate lengths counted along them are also proportionate lines. If the arc NP = arc LM, add MN to both, then arc PM = arc NL. This enables us to exchange the second and third terms in the proportion, agreeably to the process called Alternando. Also the arc PL= PN+ 2NL=ML + NL. Hence if we count from L, and call arc LM= m, arc LN= n, arc LP = p, the test of necessary direction is p = m + n. The simplest case is, when the four proportionals become three by the second and third coalescing, as if M and N run together in Q. Then if arc LQ=arc QP, we have OL: OQ= OQ: OP. If further arc PR = arc PQ, then OQ: OP = OP: OR; and so on. 4. Apply now this to the case in which the arcs PQ and QL are both quadrants. Then OQ is the mean proportional between OL(= 1) and OP=(-1). Q The received symbol for a mean proportion is V/, as in OQ =J OP. OL. Here then OQ = /(- 1. 1) = V/-1. This is only a following out of analogy with the symbol; p 0 L though, previously, </ expressed the mean proportion between numbers, or perhaps lengths, without cognizance of direction. Now, our first care must be, to inquire whether /- as a symbol of direction, has the same properties as when it operates on a pure number. First, in combining factors, the order is indifferent, as ab = ba, and a. (1) = a = 1. a. We ask, does \/- fulfil this condition? Evidently, a. /-1 = V/-1. a, each measuring the length a, directed along the perpendicular OQ. Similarly a. - b= / - b. a = b -1. a = ba/ -1. Next, repeat V-. We had OQ= OL V- 1 or -1. OL. Also OP=V /-1. OQ, because QOP = 90~,.-. OP = - 1. (V - 1. OL). But OP= - OL or - 1. OL. Evidently then V/- 1. /-1 is equivalent to - 1. This further justifies the change of / L to - /-. 5. But a new difficulty arises in adding unlike quantities, i.e. in connecting them by +. If along radii m and n we have two lengths am and bn, what meaning can we attach to am + b? This urgently needs explanation. It may seem that the symbol + (plus) receives 16 SIGN + NOW FOR TOTAL RESULT. a new sense.-Now in fact when (a + b) = zero, the + does not strictly mean addition; it really expresses a difference, not a sum; but not to embarrass generalization, we call it a sum, and say that either a or b is negative. They may mean the very same line OL estimated in opposite directions, as OL and LO. If OL mean the line as travelled from 0 to L, and LO the same as travelled from L to 0, the statement OL + LO = zero, clearly means that the total result of such travel is nothing; since the travel neutralizes itself. Thus if, instead of saying that the sum is zero, which gives only a numerical idea, I call total result zero, you will gain a geometrical idea. At this we must aim, when we deal with lines differing in direction. Evidently, if, starting from any point in the outline of a limited surface, a point travel round the circuit, until it regain its original place, we may justly say, the total result of such change is zero; and no one will suppose it to mean that the length of the circuit is zero. So if there be a triangle ABC, we may say, the total result of the travel AB + BC + CA = 0, if it be understood that each line is to be estimated in a different direction. Indeed, suppose the lengths of the three sides are c, a, b, then in the equation cm + a + b = 0, the symbol + cannot mislead us, though its sense is evidently enlarged from sum to total result. Again, since AB + BC + CA = 0 and CA =- AC, when direction is considered, we have AB + BC - AC = 0, which further justifies AB + BC = AC. The last is interpretable,-" Motion along two successive sides of a triangle yields the same total result as motion along the third side." The word resultant characterizes mechanics, but there seems no objection to adopting it in geometry also for the total result, as distinguished from the sum. 6. In fact we have unawares made a great step forward; for the symbol //- now enables us to express distance in every direction. If our parallelogram become a rectangle, and AB is the ordinary positive axis, and (as before), the lengths of AB, BC, CA are c, a, b, we have BC = a - 1 when direction is estimated, and AC = total result of c and a /- 1, or AC (an oblique line)= c + a /- 1. Since c and a are independent lengths, A i may have any direction whatever. But again, we must inquire whether the symbol +, thus extended, can be worked in the received method. First, does it fulfil the fundamental condition expressed by A +- B = B + A? Assuming (as CONFORMITY OF y/- TO ALGEBRAIC LAW. 17 we must) the doctrine of parallel straight lines, and considering any rectangle whose sides are a and c, we find that c+ aV/-1 = the diagonal = a /-1 +c from the opposite sides. Next, does it fulfil the condition h(a + B) = hA + hB? p The doctrine of similar triangles at once affirms it. Let OM=x, MP =y, perpendicular to it; join OP, then if OM is the positive axis, and y expresses mere m M o length, we write OPor OM + MP= x+ Y/-1. Next, along OM take Om = hx; that is 1 h = x: Om (whether h is linear or numerical). Erect mp perpendicular to Om and meeting OP in p. Then by similar triangles, mp = h. y (in length) and Op = h. OP. In this h. OP = Op we have supposed h to be numerical. Also OP is equivalent to x + /- 1. y, and Op to Om+ np V/- 1 or hx + V/- 1. hy, that is, h (x + /- ly) = hx + - 1. hy, just as if V/-1 were nuinerical. If further we change h from a mere number to a positive length, it affects every term of the last in the same ratio, and leaves equivalence as before. If we have proved generally that with any factor h (provided it be counted along the positive axis) the product h (x + V/- 1. y) is equivalent to hx + V/-1. hy, the same is virtually proved, if h be changed into h, that is, if the numerical h be computed along an m-axis. For we may transform our hypothesis, by choosing the m-axis as positive. If hereby x, y change to x', y', we obtain a result the same inform as the previous result, and x', y' remain quite as general as were the x, y. Thus we may write ^ (. (+ V- ly) h-n + v-1.h,,,y. After this, we can change the oblique h,, into a + /- 1. b, where a, b are along the positive axis. Now if the m-axis be perpendicular to the positive, we may write simply hm = k /- 1, where k is along the positive axis. Then V/-1. hmy = /-1. / -1. ky, and since each V- 1 N. 2 18 CONFORMITY OF / - TO ALGEBRAIC LAW. denotes revolution of the ky through 90~, the 4/- 1. -/-1 shews revolution through 180~, or is equivalent to the symbol -. Thus h,(x + /- ly) = 4-1. k(x + V- ly) = V/-l.k + /-1.k. Â/-1.y = -1 lkx - ky exactly as if V/-1 were numerical. Evidently then the same holds good in multiplying out (h + /- lk) into (x + 4- ly). THEOREM. If A + B V- 1 C + D V- 1, this implies two equalities, viz. A = C and B = D. The geometrical proof appeals at once to the eye. If OA be the positive axis, and the binomials are denoted by OP and OQ, Q P viz. A +V-1B = OP and C + V-1D = OQ, we do not account OP= OQ until they have the same direction, as well as the same length. This requires Q to coincide A N I v o with P. Of course then, if PM, QN are dropt perpendicular to OA we have OM = A, PM = B /- 1, ON = C, QSV=D /-1, and as soon as OP = OQ in our hypothesis, P coincides with Q, therefore also M with N. That is A = C and B = D. The reader will now see the geometrical meaning of the "imaginary roots" (so called) of a quadratic equation. As a very simple case, take first x2 - 16x + 63 = 0, which yieldsx = 8 + 1. Here both roots lie along the positive axis. But change 63 to 65, then X2 - 16x + 65 = 0, whence x = 8 + - 1. In the latter the two roots are equal radii drawn from the origin at equal angles on opposite sides, radii which terminate where the coordinate along the axis is 8, and the transverse coordinate is + 1. TRACT III. ON FACTORIALS. SUPPLEMENT II. Extension of the Binomial Theorem. 1. THE following appeared in Cauchy's elementary treatise, as early, I think, as 1825, but without the new Factorial Notation, which adds much to its simplicity. Boole writes x(2) for x(x -1), x(3) for x (x- 1) (x- 2), and x(" for x (x- 1) (x- 2)...(x- n + 1), whence (+') = x(. (x -n). Better still it is, to place the exponent in a half-oval, since a parenthesis ought to be ad libitum. I propose XI-c, tj: which are quite distinctive. Then the Binomial Theorem is X X 2 XS (1+)=l +-) 1n. 1 n. 2+n-.1 +... + In this notation 1. 2.3.4... n or n (n - 1)... 2.1 is n-. Gudermann for this has n'; but (n - 1)' is less striking to the eye than n, In - 1 introduced by the late Professor Jarrett. This exhibits in the Binomial Theorem its general term, by (1 + x) =l+n. - +...+ +.. +; of course x,1, is equivalent to simple x. The Exponent (of a power) is already distinguished from an Index. In a Factorial x for x (x- 1)(x- 2)...(x - r-1) one may call r (which must be integer) the Numero, as stating the number of factors. 20 EXPANSION OF (x+ h)&. 2. If m, n, p be all positive integers, and p = m + n, then (1 + x)y. (1 + x)n = (1 + ), or, in condensed expansion, {l+2m ~X }.{l+.Xr + x ={1+ 0P ~x This equation being of the (m + n)th or pth degree of x, and being true for values of x indefinite in number (therefore in more than p values), must be equal term by term for every power of x. Now when we multiply any two such series, (1 + Mx + M22 + M3x3 + etc.) by (1 + N1x + N2X2 + etc.), we have a product of the same form 1 + P1x + P2x2 + P3 +... by the routine of multiplication, in which P,=M1+N1; P2=M2+MNl +N2; P3,= M3++MN+M1N2+ N3; and the law of the indices is so visible, that we get generally P, = M + MIlN, + M_-IN +......+ MN + N,. This being true for all series of this form may be applied to the three series (1 + x)", (1 + x)", (1 + x)P, and at once it yields to us the result Ir r-i r-i 2 r-3 3 r p,- _ m - m^- n m- 2 n.- ml23 nL3 n. I _ + ir-1 'l+ I-2 * I r-3 ' 13 + '* + an equation true for more values of m and n than are counted by the integer r, therefore it is true also when m and n are arbitrary and fractional. Write x for m, h for n, and x + h for p, and you have an extension of the ordinary (x + h)y. For, this latter may be written (X + h)r xr x - h Xr-2 h 2 x -3 h h [ - + + + + - L _r I.r-l + lr- '2+ Ir-.3' 13 1 Ir; with exponents replacing the numeros of the preceding. NOTE. The reader must carefully observe in that which follows, that the upper index of P, Q, A, B, C, is not an exponent. POWERS IN SERIES OF FACTORIALS. 21 Powers in Series of Factorials. 3. Since x- = x (x -1) = x2 - c, of which the general law is C (x - r) = xr, conversely xa2 = x + x. Again xL = x ( - 1) (x - 2) = x3 - 3x2 + 2x,.. X3= X, + 3x2- 2x. But 3X2= 3x2 + 3x,.. x3 = 3 + 3x- + 3x. Evidently we can thus in succession obtain 4,.x5,... and generally x" in series of n-, 1,... -, a. Since only xa- contains a", its coefficient must be 1. In general, with unknown coefficients P, dependent on n, but not involving x, we may write n =pn n + pn-1 xn + pr,, 2... r+ P'l X +.-PLi *(' a"'= P~~a'+P Z~c/+P 2a'+... +P 2_2-~P1 +Pl-lx...(a). Here the lower index denotes the place of the term in the series; the sum of upper and lower index = n, the exponent of x"; and the upper index is the same as the numero of its tern. We have also seen that PO= 1, whatever n may be. It remains to calculate P.-' Multiply the left member of (a) by x, and on the right multiply the successive terms by the equivalents of x, viz. (x - ) +, ( - n + 1) + (n - 1), ( - n + 2) +(n-2), etc., and apply to each term the formula xa. (x - r) = x. Then æn+l p= Pn n+l D- + p + Pn-P2n 1 + + - + -i + n P. * + (n-1), -. - - + ( - 2) p-2 (-2) +... (b). 2 + 1PI + 2P 2 a'-1 + 1P-1 xJ But if in (a) we write (n + 1) for n, we have 11 = P1'_ + pPn + P P + x.. + PrP 1 +P... + (c), and we cannot be wrong in identifying (b) and (c); that is, in equating the coefficients belonging to every particular numero (r). At the right hand end Pl== 1P'_, coefficients of x; i.e. since when n = 2, x2 = xL + X, so that P=1, * 7^i 1 M 1 - 22 INVESTIGATION OF TABLE and universally P-= 1, just as Po= 1. Also in general we find pr r r-1 2n+l-r = -r P Pn+;l-r) otherwise, P= rP_ + P;1, if p= + l-r. This enables us to fill in the vacancies of a table, beginning from 1 1 1 Pi P3 P4 2 2 2p P2 P3 P4 3 3 p 1 P2 P3 P4 one horizontal row and one vertical, each consisting of units. Each P is computed from the P above it, multiplied by its upper index (which is the number of its column) +its companion to the left in the same row. Thus to form the second row from 2.(1)+1 3.(1)+3 4.(1)+6 5.(1)+10 1 =3 =6 =10 =15 Evidently this second row is 1; 1+2; 1+2+3; 1+2+3+4;... of which the general term is PR='n.(n+ 1). Similarly from the second row we form the third, workingfrom left to right, and the law is manifest. 1 3 6 10 15 21 2.(3)+1 3.(6)+7 4.(10)+25 5.(15)+65 6.(21)+140 1 =7 2 = =65 =140 =266 Hence a table to any extent required can be made,`such as is here presented. TO CONVERT POWERS INTO FACTORIALS BY P. 23 pi p2 p3 p4 p5 p6 0 1 1 i 1 1 1 1 1 3 6 10 15 21 2 1 7 25 65 140 266 3 1 15 90 350 1050 2646 4 1 31 301 1701 6951 5 1 63 966 7770 42,525 6 1 127 3025 34,105 246,730 7 1 255 9330 When this table is used solely to evaluate the coefficients of (a), the indices of Po, Pl"-l, pn-2... warn us that the numbers will be taken out diagonally. Thus for x6 we take out (beginning from P' at the top on right hand) 1, 15, 65, 90, 31, 1. It will be observed that the second column is 2 - 1, 22 - 1, 23 - 1 and in general P' = 29+l- 1. Such is the Table of Direct Factorials. The letter P being almost appropriated for the Legendrian functions, I see an advantage in superseding Pn by |. Necessarily n and r are both integers, n index of the column, r + 1 index of the row. Collect the results for P P= 1; Pl= 1; Pt= n(n+1); P2 = -' 1; and in general Pr = rP/_ + P1-l, which yield identically x =[ ~. x2 + xI+." +| 4etc. Factorials in Series of Powers. 4. This is a mere problem of common multiplication, x - x (x-1) (x- 2)...(x- n + 1), yet the factors being special and their combinations often recurring, 24 INVESTIGATION OF TABLE the work of one computer may avail for many after him. We may assume nXe = -Qo QW1n-'1 + Qrn-nx'2 - etc.... + Q..... (a); where, as before, obviously Q= 1. Also dividing by x, and then making x = O, you have Ql,_= - 1.2.3...,(n-l)= I n-1. Since (x - n) x- = x, multiply (a) by x - n,.x+ = Qno Q n -ln x + Q2z-2x- - etc.... + Q,_1 2 b) - nQ it + nQ1'-ln" - etc... + nQ,2e2 -T nQ-n_ ) Also in (a) write n + 1 for n; ~+ = Q"ln+l - Qxn + Qn-xl - etc......(c) But (b) and (c) ought to be identical. We have anticipated the remarks that Q +1 = Q= 1, and Q-= nQ _1, inasmuch as Qn=1.2. 3...n; = In. But generally, Q-r = nQ-r + Q-r-1; or, if n - r =m, Q = (m + r) Qr + Q.+. As before, this enables us to continue the table, when the first row and first column are known. To compare our formula with that of the first table, we may write it Q;= (r + p-1) Q;, + Q;-'. In fact, the first row is unity as before. The first column is 1, 1, 1.2, 1.2.3, 1.2.3.4, when r 0, Q = mQ+ -+ Q'also in the former table, when n = r, P = rP; + P-'. Hence the second row is the same in the new table as in the old. To compute the third row from the second: 1 3 6 10 15 21 4.(3)+1. 2 4.(6)+11 5.(10)+35 6.(15)+85 Qp=(r+p-l) Q 1.,2 1=11 =35 = 85 = 175 -+ Qr TO TURN BACK FACTORIALS INTO POWERS BY Q. 25 The multiplier r + (p -) combining upper and lower index of its Q distinguishes the Q table from the P table: thus 1 1 35 85 1.2.3 4.(11)+1.2.3 5.(35)+50 6.(85) + 225 =50 =225 = 735 1.2.3.4 5.(50)+2.3.4 6.(225)+274 etc. = 274 = 1624 or indeed Q1 = (r + p) Q;+1+ Q;. Inverse Factorials. Q1 Q2 Q3 Q4 Q5 Q6 Q7 0 1 i 1 I 1 1 1 1 1 3 6 10 15 21 2 1.2 11 35 85 175 3 1.2.3 50 225 735 4 14 274 1624 6769 5 15 1764 13,132 6 16 These too are used diagonally for xe. Thus xL'7 = I - 21x6 + 175x5- 735x4 + 1624x - 1764x2 + 16. x. Again it seems better to supersede Qr by, then xo = xM xn _ [+ X-2 + etc. 5. Even in Arithmetic we are driven upon "recurring decimals," and learn that an infinite series may tend to a unique finite limit. Nor can Elementary Algebra fail to recognize, from i - xa = 1 +x+ + +,.. x"i-~ 1-1x: 26 WITH A NEGATIVE NUMERO that when x is numerically less than 1, with n indefinitely increasing, the series 1 + x + 2 +... + xn tends to the limit - 1-x After this it quickly follows (by Cauchy's process now perhaps universal), from Binomial Theorem with n positive integer, that (1 + )" with n infinite, has for limit l + +- + -- etc. = 27182818... 1 1.2 1.2.3 which we call e, and that (1 + ) has for limit n a a1 a2 1++ + - +t.... i 1.2 which also = ae. On this we need not here dwell: but it will presently be assumed. Now let us propound Factorials with.Negative Numero. 6. Analogy suggests to define xZ as meaning (+: of course x (x + 1) x,- will be identical with x-'. Then xæ3 will stand for [(x1) (x+ 2)]-', and x< for [x(x + 1) ( + 2)... (x +n - 1)]-. Hence x-Z' = (x + n)-'. x Now <2 = ( X + 1)X I, and when x is > 1, we have, in descending powers x- = x-2 - x-3 + a-4- -5 + etc. Also x~ = (x + 3)-1. x2 of the two factors here on the right, each can take the form of a series descending in powers of x. So then their product, the equivalent of x3. By like reasoning we claim a right to assume with coefficients independent of x, x-,n = A'nx-n-l _ - - + Ax — 3 etc.......... (a), and our task is to discover the coefficients when n is given. THE TABLE OF P SUFFICES. 27 First make n = 1, x.-2 = A ~ - A X-3 + A -4- A3-5 + etc. But from the series already obtained for xe we see that every coefficient of the last is 1; or in general A = 1. This gives the first column of our table. Also universally AO obviously = 1; which fixes the first row of the table. Next, multiply equation (a) by (x + n), and you get X n _ n-n -n-1A + Anx -etc.... A xn-"-.. + n-n-l -_. A x —2+ etc.... TnA x-"-r. But in (a) we may write n for n + 1, which gives X n _ X-n _ An- + i.. + An-l1 x-?n-r +. ( c) = x-"- A+..._+_An TF..............(c). Now (b) and (c) niust be identical, hence A n= n+ Ain-= nAo + A-1, and generally An= nAn + Ar-l But this is exactly the law of P in Art. 3, only there we had r, p for what are here n, r. Now as the first row and first column are here, as there, unity, and the law of continuation the same, the whole table will be the same, and we may write P of Art. 3 in place of this A. Thus we get -n =-n _- n-l -n- 1 n -I nl -t- + etc. — x = xn - P x-'X" + Pr-o-e - P_ a + etc.... with the same values of P as before. But in the last equation we no longer take the P's diagonally, but vertically, down the column, as the same upper index (n - 1) above every P denotes; thus x3- = x3 - 3x-4 + 7x-S - 15x-6 + 31-7 - etc.... To verify, multiply by x + 2. But for convergence x ought to exceed 2. So 4 = x- ~ - 6x-3 + 25x-4 - 90x-5 + etc.... and for convergence, x ought to exceed 3. Evidently in the series for x., x ought to exceed (n - 1). 7. Assume now the Inverse Problem, to develop x-n in series of Factorials.. With unknown coefficients B independent of x, we start from x-'1-l - x - +. x 1- l 32 +" +..... '....... (a). 28 FOR THE INVERSE PROBLEM, Q SUFFICES. Multiply the lefthand by x, and the successive terms on the right by the equivalents of x, viz. (x +n)-n, ( + + 1) - (n + 1), (x + n + 2)-(n + 2), etc. Observe that (x +p) xI —1) =(-; * nX-n = Z() + B n1.- +. ~x-"~=xo:<++B'.xe-+... + B x,. +..,. - n. ' (n + 1) B~. x, _ —... (n r). BT-1 _... But in (a) we may write n for n +1, which gives x- B +,-n- + B' -lx + + Br... + x-, +....... (). Identify (b) with (c),.-. B= n - Bl-; and generally Bn = (n + r - 1) B_ + BT-; the same formula as for Q in Art. 4, only n and r here standing for what there was r and p. Also since Bo= 1, the top row is unity, here as there. We may further prove that the first column of our B's is the same as the first column of the Q's. For t-2 = x + B' x + B x +... &c Multiply by x on the left, also by (x + 1) - 1, (x + 2) - 2, (x + 3) - 3, for the successive terms on the right; then x-l = x + B\x2^ + B1x +... -1 2- _ 2Bi - 3-3B e 4-... Obviously x' = x_; and the other terms must annihilate themselves, making B-1 = 0; B - 2B= 0; B3- 3B = 0; etc., or Bi =l1, B= 2Bl=1.2; B = 3B =1.2.3, and B = 1. 2.3... (n-1) n. Thus BI= Q, BI= Q', etc. exact. Therefore the B table is the same as the Q table throughout. Here also we take the Q's vertically, to obtain axn in series of factorials as X4 = X.4 + 6xc. + 35x + 225xz_7 + etc. In general x-" = x,: + Q-lx — Qn-2 -... + QQ-l - + + etc.; but special inquiry is needed concerning convergence. Apparently it converges more rapidly than x-2 + (X + 1)-2 + (x + 2)-2 + etc. NEW. PROBLEM. 29 {x x2 x3 $4 ) 8. To develope (e - 1)" or + + + + etc. in powers of x". We may assume for this the form Mxn + M.nxl +... + Mx + etc., where M, manifestly = 1, and if r is less than n, 2 = O. Now by the Binomial Theorem where n is a positive integer n - ( _ l)n = en a(n-l)x +.e6(n-2) - etc. n$ T-l, 1 |ec 1 out of which we have to pick up the partial coefficients of x which make up Mp observing that e = l+P+ -+ + etc. in which we have to make p successively n, n- 1, n - 2,... When we make p = n, the only term that here concerns us is r. When Ir p=n-1,weget -. Whenp=n-2,wehave n2 (n-2). x + 12 * r and so on. All have Ir in the denominator. Hence M. r= n' — (n - 1) + n-. (n- 2)- etc.... to n terms, of which the last is + 12. -1' Let N be to (n + 1) what M is to n; then, with r the same for both, n+ ( +r =(n 1) (n) + +~c (7' 1 ) (n -1)' -( 1. (n-2)' +... to (n + 1) terms. Add this to the preceding, coupling every pair that with the same value of p has (n - p)r as factor;.*. (M1 + Nr) = (n + l)- 1 ] (n)' + [n21- 1]. (n~-l) _ [WT]n.n- I ((n - 2)' [n+ 1 1n n '72 n-2 30 COMPARE Mll + N, WITH Nr+l n, +1 n -+ n-1 n+l n-2 Observe that -1 = n, - -1 == i 2 2' 3 3 and so on;.* r. ( I- N) = (n + 1)" ' ()r +Î n2 (n - 1)0 - etc., or, in shorter notation, = (n + )y n. (n) (n 1) n. (n - 2)2+ etc. to (n + 1) terms. But in N7 change r to r + 1 and you have |r + 1i. = ( + r 1 ( + + + I. (n - - n+' In nl. (n -2)+2 + etc.... to (n + 1) terms =(n +) {(+l) r- (n+ (n-)r+l + 1 (n-i)+n + et.... = (n + 1) ((n + r - (n - n.n-1.n-2. n- (n - 2/-,+etc.} = (n + 1). Ir (M, + ), from above. Divide by (n + 1). r1; n+î r+1 (a). * n+ i r + l............... Our r always exceeds n. We simplify the notation somewhat by assuming as coefficients after dividing by x", 1( - i\n C1 C n2 Cn xS + + 2 + \: n+l n+1 n+ 2 n+l.n+2n+3 Cn Xp + 4 +4 etc. + 1. n +-2... n+p Multiply by x". The general term becomes C(. n +p (n+l - -1t '~ t' _ o' t% ~, 'l \ ([n +.,,, THE TABLE OF P AGAIN SUFFICES. 31 Put n + p = r, to identify this term with our previous Mr. x; Cn * M - = _______ Lr-n * r n+1. n.2. n +3... (r-1).r' Mn.+l so that NT= - r-n-l with one less factor in (lenominanr +2.n+3...r-l.r tor: but when both n and r in M,. are increased by 1, r - n undergoes Gn+l no change; and N +, = n+2.n+3...r-.r. r + Introduce these values into (a) and multiply by (n + 1) (n + 2)... r. (r + 1),.. Cr+ = Cr + (n + 1). C'-...... (b). Change r- n to p and n +1 to n,.. CP = Cp-+ nCi; the same law as for the P's in Art. 3. Also when n = 1, Ex 1 O )C2 $r - =1+ l + 2 +... -t + etc. \(~ ~ x 1-1.2. 3 + 1.2. 1..(r+l) But we assumed as equivalent (1 C Cx2 GCX3 c1_ r i _ ++ _ r + _ ' 2 2.3 2.3.4 * 2.3...(r+1) which identifies C1 with 1, for all values of r. Just so P' = 1 in its first column. Also evidently C =l1; just as Po=l, in the whole first row. Thus, with first column and first row identical with those of P and the same law of continuation, the whole table is the same. Finally then we obtain 1 + n + n + 12 n + n + 2. ____ L P^. -2 ______.____3 - etc. (c )~ n+1l n+l.+i+2 n+I.n+2.n+3+etc' (c) in which the increasing numerators are pulled down by increasing denominators. The general term may be written n2. P. x^, or equally Pnl. oe" (n +p) ' [The course of analysis here pursued forestalls that of A. or to the learner.] Thus (e - 1 L ++ + ++ etc x +1 n +1.n n 2n+. n+ 2. n +3etc' 32 PROBLEM OF LOG (1 +y). 9. To investigate log (1 + x) in series, with no aid from (1 + x)" except in the elementary case of n being a positive integer, and no aid from the Higher Calculus. Let y= - 1, then x = log (1 + y). Also x x2 X3 y- ++ - + — 3+ etc.... by elementary algebra. From the last we see that for minute values of x, as a first approximation, y = x, and x2 = y2. As a second, y = x + xo2, = x + y',.'. x=y- y2. Whatever the series in powers of x, x can be thus reverted into powers of y, at least within certain limits. Hence we may assume with unknown coefficients called A, x or log (1 + y) = y-A2y2 + A3y -A4y4 + etc....... (a), which (with the appropriate numerical values of A2A3A4...) will be an identical equation. Consequently, writing y + z for y, log ( +y+) = (y + ) - A(y + )2+A (y + ) + etc.... (). Subtract (a) from (b) developing the powers (y + )2, (y + )3, (y + )4... then log (1 + y + ) - log (1 + y) = - A, (2yz + 2) + A. (3y2z + 3yz2 + Z3) - A, (4y3z + 6y2z2 + 4yz2 + z4) + etc.... (c). But the left hand = log. ++ or lo 1 + ) which again, by (a), if we write for y, has for equivalent + y ( A)-.*(.+) + A A )-etc.... of which the first term alone contains the simple power of z, and there its whole coefficient is ( 1). This then must be the sum of the partial coefficients of z found in (c). That is = 1 - 2Ay + 3Ay2 - 4A4y3 + etc. But 1y+ ec n y n erically ess t 1. But =1 -+ y2 - y2 + etc. when y is ntunericaUly less thant 1. + y PROBLEM OF [log 1 + X12]. 33 Hence 2A =1, 3A =1, 44 = 1,... nA 1, or in general A,, Finally then, log (1 + y) = y - ly2 + y3-_ y4 + etc... while y < 1. N.B. This furnishes the means of computing logarithms in Elementary Algebra, before knowing the Binomial Theorem with negative or fractional exponent. If Differentiation or "Derivation," as far as f (x) = x, 0'(x) =-nx'", be admitted, this equation at once proves that wlien b (z) means log (z), ' (z) -: a process which eases the next Article. 10. PROBLEM. To develop [log. 1 -+ x] in a series of powers of x. That this is possible, when x2 is < 1, the preceding Article shows. We may then assume, with unknown coefficients X', XM, k... depending on n alone, where the upper index is not an exponent, -x,+l x1 X2n+ z=(log. 1 + x)" = " + - - etc.... n+1 n+i.n+2 Àn. n+r + n+.- +etc....... (a). n+i. n+2....+r+ r If log + x=, z=u, dz=nu1-du, and du= d. [It is i +x hardly worth while to disguise' Diffetentials by a more elaborate and tedious Algebra.] Differentiate both sides of (a), then drop the common factor dx: hereby, n (log 1 + x)>-'= n -' +l - X. w n X x e+l 1+x + 1 etc... + __X~. x" + etc n+.n+2 (n+r-1 n + 1. n + 2... (n + r- ) etc. Multiply by 1 + x, and divide by n, n~ = - x" X '. x"+l n n.n+l 1.~.(lor 1 + x)"-. = x"'l -_ X, _ +, l - etc. X~. x"++rl+n+r-et...... (b). n.n+1... (n+r- 1) + + o,; -À +etc. T - -p +- - ~etc. n n +. n +1... (n+ r- 2) N. 3 É34 THE LAW OF Q SUFFICES. But writing (n - 1) for n in equation (a) we get (log 1 + x)"- = x- L _ + AX 2 -etc.... n n.n+1 rn-l &n+r-l n. n +... +r-l1) + etc......(c). Identifying (b) with (c), we obtain Xl=: n +X -1; and generally X: = (n + r - 1) Xr-l + X-1 or X+i=l (n + r) X7 + Xr+, the same law as of Q in Art. 4. Also evidently Xo =1 whatever n may be, as Q =1. Thus the top row is 1. Again by (a) making n = 1, 12 Ml x3 X134 log(l+x)=x- + - +etc. 2 ++etc. But this is known to be x - x2 + x3 - x 4 + etc., whence X:= 1, X = 2, = 2., X=2. 3. 4,...just as Q= In. Thus the first column also of X agrees with that of Q. In short then, the two tables are the same. Finally: Clog 1 + X)" = x" - Qx+ Q. + etc., ( +) n + -. n + 2 n +l.n +2.n + 3 with coefficients already known. The analogy to the series of Art. 8 deserves notice. (log.1+X n n 1 X n + ( n+l n+l.n+2 n+ - -.+2.- + etc. x ) l n+l. n+ 2 n+ L. n + 2. +3 TRACT IV. ON SUPERLINEARS. SOME AIpology may seem needful froin me, since Dr Todhunter in lis volume on Higher Algebraic Equations has treated the same subject under the name of "Determinants." Of course I do not pretend to add to him, nor indeed he to Mr Spottiswoode, who carries off all merit on this subject. I read the details with much admiration as treated by the latter, but found his notation by accents very dazzling to the eye, in so much as to make it hard to know by sweeping over half a page, what was the meaning of the formula presented to one. Also I found the chief strain in argument and chief liability to error, to turn upon the question, whether this or that resulting term would require a plus or a minus. By reasoning from linear functions in my own way, though less direct, I was able to avoid this danger and lessen fatigue to my brain. When I exchanged words with my then colleague in University College, London, the late Professor De Morgan, on the question, why this topic was not admitted into common Algebra, he replied, that it was too difficult for beginners. I have since thought that he might not have so judged, if some of the arguments were otherwise treated; and in fact I have found, that some whom I supposed to speak with authority thought my slight change of method easier to learners. At the same time I must add, that on the very rare occasions in which I have tried to teach an elementary class of mathematics, a mode of reasoning which to one pupil was easier, to another seemed less satisfactory. Perhaps every teacher ought to have " two strings to his bow." 3-2 36 BILINEAR TABLET.P 1. Equations of the first degree are called simple, but when three or more letters (x, y, z...) are involved, complexity arises with much danger of error, even when there is no difficulty of principle. To solve two equations of the form ax + b,y = c, ax + by =c is always the same process. If we could always certainly remember the solutions clb - cbl c2a1 - cla2 a b -ab' alb - aab and never confound the indices, nor mistake between + and -, this alone might have much value. The modern method, due eminently to the genius of William Spottiswoode, is quite adapted to Elementary Algebra; but its vast range of utility cannot there be guessed. First study the denominator D = ab2 -a2b1. It arises from the left-hand of the equation (ax+by= c,, in which the four letters a2,x + b2y = c2 stand in square, as a1 b1. Here D is the difference of the two a2 b2 productsformed diagonally, and the diagonal which slopes downward front left to right is accepted as the positive diagonal. This is a cardinal point. Remember it, and you will not go wrong on + and -. Understand then, that in square with vertical sides MN I means MQ-PN. P Q Of course then, so does M P, which exchanges rows into columns. N Q But if you change the order of the rows, or the order of the colurnns, into P Q or M you change the result to PN - NQ, i. e. you MN Q P change D to - D. After this is fixed in the mind, it is easy to remember the common denominator above, viz. alb2- a2bl, in the form al b. Call a2 b2 it C. Then the numerator of x is obtained from C by changing the column ala2 into the column c1c2, making A = c1 b c2 b2 So B for the numerator of y is found by changing the column blb2 into c,c, yielding B = a, c, a, c2 A B Finally, x =7, y= A without mistake. 0j, y = LINEAR FUNCTIONS. 37 Observe, if the equations be presented in the form aLx + by+ c=0; ax + b2y+c= 0; this is equivalent to changing the signs of c1 and c2, which does not affect C, but exactly reverses the signs of A and B. Previously we had (.: A)= (1: C)= (y B), C2 b2 a2 c2 a2 b2 or ($:' 1) =(A: B' C)= C2 b, | | a C~ 1 I a b But from the two new equations ax + bly + cz = 0) you get a2x + b2Y + c2z = 0 b1 c, c, a, a, b,: y: == 1 1: 1: l / x':y:2= |2 c2 c2 a2 2 b2 in circular order. We may also present the solution as follows: x y z b c c1 c a a b, b2 c2 c2 a2 a2 b2 2. Simple equations are often called Linear, by a geometrical metaphor. If a quantity u is so dependent on x, y, z... that however the values of these may vary, yet always u = ax + by + cz +... (where a, b, c... are numerical), then u is called a linear function of x, y, z... its constituents. [One might have expected u to be called a dependent or a resultant: but for mysterious reasons of their own, the French have adopted the strange word function; and it cannot now be altered.] It is convenient now to set forth a few properties of linear functions. We here suppose the function to have no absolute (constant) term. I. To multiply every constituent by any number (m), multiplies the function by that number. [For, if u = ax + by + cz +..., then mu = amx + bmy + cmz +....] II. If two linear functions have the same number of constituents and these have the same coefficients [as U= ax + by + cz, :3.8 LEMMAS. ON LINEAR FUNCTIONS. U = ax1 + by1 + czl]; you will add the functions if youth join the constituents in pairs [for here U+ U = a (x + x1) + b (y + y,) + c (z + z); whatever the number of constituents]. 3. Observing now that a or aN - bM is a linear function b N of N and M, we see that to multiply a column MNT by m multiplies the function by mi Or a, mil = m a M. b, mN b N The same is true if we multiply a row; for we may regard a and i as the variables and b and N as constant; then ma, mM =m a M. b, N bN This leads to the remark, that our function ought to be called superlinear rather than linear; for it is open to us to suppose constituents alternately constant or variable. 4. Next, by making a colurnn or row binomial, we can sometimes blend two superlinear tablets into one. Thus Ax + Cx, B y D Y having the second column the same, yield, (Ay - Bx)+(Cy-Dx)=(A + C)y-(B+ D)x = A + C, x B+D, y Here the column which was in both, remains as before, but the other columns are added and make a binomial. Evidently the same process holds, if a row, instead of a column, is the same in both. Conversely, when a given column is binomial, we can resolve the tablet into two tablets; and when each colunin is binomial, we can resolve the tablets into four. Thus A+x, C+z = A, C+z + x, C+z B+y, D+v B, D+v y, D+v by a first process. By a second, each of these tablets becomes two, giving as result, A C +lA z + x C + x z B D B v yD yv VANISHING BILINEAR. 39 5. THEOREM. If one column (or row) is identical with the other, the tablet = zero. For obviously A A, by definition, is AB - BA or zero. B B COR. Equally the tablet = zero, if one column (or row) be proportional to the other. Thus if A B= C: D, this proportion yields AD= BC; hence A C or A B, meaning AD - BC, vanishes. B D CD 6. This sometimes usefully simplifies a tablet. Thus A + mC, C B +mD, D is resolvable by Art. 4 into A C + m C, C. But the second B D mD, D tablet is zero, because its two columns are proportional. Hence the given tablet simply A C. B D COR. Hence an important inference. The value of a tablet is not changed if to one column (or row) you make addition proportional to the other column (or row); nor again if you subtract instead of adding. Further, if in the original equations of Art. 1, the absolute terms c, c2 become zero, the solution for x and y is simply x = O and y =. This is indeed the only general solution, unless also the O O denominator vanish; which makes x = O, y = deciding nothing as to the value of x or y. In that case we can equate the two values of Y; viz. = - y_a= Conversely, this shows a b2- a2b, x' b1 x b' b equivalent to a1 b = O. The last is the condition which provides a2 b2 that the two equations shall be mutually consistent, though x and y do not vanish. It results from eliminating x and y, leaving their value arbitrary. 40 TRILINEAR TABLETS. 7. Begin from the problem of three simple equations, f ax + b1y + cz = 0 xa2 + b2y + c2z = O These present three equations to be fulfilled by only two disposable quantities, viz. - and Y, if they are all divided by z. The three z X equations are not certainly self-consistent. If x, y, z be eliminated, an equation of condition will remain. Our first business is to investigate it. From the two first equations treated as at the end of Art. 1, but abandoning circular order, we have bi1c1 a. C1 ai b1 yW:1:8 bZ c' a c, a2 b in which - a. c, now replaces + c ai of Art. 1. a c~ c~ a, a2 C2 C2 2 In the third given equation, substitute for xyz the three quantities now proved proportional to them, and you get (1) ai b c b +-C3 ac, b = 0. b2 c2 a2 aa2 b2 Call it V =0. Then V3 is linear in a,,b2c. This is the condition that the three equations shall be compatible. It is seen to result from eliminating x, y, z. Professor De Morgan wished to call these tablets Eliminants. Why Gauss entitled them Determinants, no one explains. Spottiswoode apparently introduced the excellent notation a, bi ci Then V. is linear in a,, b., c3, that is, (2) V] = a2 b2 c0 in its third row. Hence also in any row. a3 b3 c3 V3 is superlinear of the third order. for the value written above. The coefficient of a3 is obtained in (1) by obliterating the constituents as last written that are in the same column or row as a,, thus reducing VY to... b1 c:... b2 C2...... then we see the tablet by whieli a3 must be multiplied. EXPANSION OF A TRILINEAR. 41 The sarne process is used with b3 and c3, producing al... c1 a, b,...... c and a b2..... b3...... c.. C Finally the signs of the terms of V3 are alternate + - -, just as in the bilinear TV = ab2 - a2b1. Evidently V3 is formed of six terms, three positive and three negative; each term having three factors, but in no term is any factor combined with another of its own row or its own column. In c 1 a b1 the term a1b2c3 (the diagonal sloping down from a2 b2 left to right) is positive as before. Thus when V3 is given in contracted form, we can expand it into three apparent binomials. 8. In the three given equations you may exchange the position of x and y; then by eliminating y, x, z you obtain b, a, c, U3 = b2 a2 c2 =0; b3 a3 c, but you cannot infer that U3 = V. In fact, to exchange the a column with the b column reverses the sign of a, b1. Thus it changes VY into a2 b2 b3 a, c, -a3 bl cl +c3 b1 a. a c2 b 2 C2 b2 a2 That is, to exchange the first and second columns just reverses the sign of V3. The same effect follows from exchanging any two contiguous columns or rows. Thus generally AD G D A G D G A B E H= E H B =+ E HB G F J F C J F J C A D G BE H BE H Again, B E H=- A D G =+C F J C F J C F J A DG 42 EXCHANGE OF COLUMNS OR ROWS. Observe, that if three binomials of V3 are expressed by ma8 + na2 + pal the multipliers m, n, p contain nothing of the column a1, a2, a3, therefore V3 is linear in these three constituents. Evidently it is linear in regard to any column; as we before saw, as to any row. 9. THEOREM. Further, I say, To exchange rows and columns does not alter the value of V3. In proof, multiply the three equations given in Art. 7, by disposable numbers m, n, p, and so assume m, n, p that when the three products are added together the coefficients of y and z may vanish. There will remain (mal + na, + pa,) x = 0, and as we do not admit x=0, we have three equations connecting m, n,p; viz. alm + ban + abp = O bm m~ b2n+b8p0 P-, c1m + c2n + Cp = OJ and that these may be compatible, we need a, a, a, 83= b1 b b =, C1 C2 c, by eliminating m, n, p. Here S8 is nothing but V3 with rows changed to columns and columns to rows, retaining the same positive diagonal alb2c3. Every learner will easily fn lined by developing and S that they are identical: but there is an advantage here in general argument applicable to higher orders. S3= 0 and 13 = 0, being each a condition of compatibility of the previous equations, must contain the same relation of the constituents. S. is a linear function of its column alblc,; so is V, a linear function of its row a1b1c,. But S= =0, V = 0 being derivable one froin the other, there is no possible relation but S3 = iV. in which,u must be free from albc1. But the same arguments will prove that u is free from a2b2c2, also from a8bsc,. Therefore pu is wholly numerical. Make b = 0, c1 = 0 and this will not affect ~p. But this makes V3=a, bc2 and S= a, b, b, b c3 c2 But that the two minor tablets are identical was implied in their definition. Hence, on this assumption for b, and c,, we find V3 = S. or, =1. This then is the universal value of p, or Y = S in all cases. Q. E. D. ON A SUPPLEMENT VANISHING. 43 10. In the developed value of Vf (Art. 7) if b= 0 and c = 0, you get simply Y8 = a8 b c which does not contain a1 or a2. These b2 c2 two constituents are made wholly inefficient by the vanishing of b8 and c8, and may be changed to zero. Thus a1 b c, O bc. So a b c, a 0 0 a2 bc == O b2 c-2 = 2 - — 2 2 0O b c O b c, a3 0 0a 0 0 o3 C When a major square is divided into two minor squares and two complemental rectangles, the vanishing of one rectangle obliterates the other, and the greatest tablet has only the two squares for its factors. 11. Since V8 is a linear function of every row and of every column, we can argue as in the Second Order or Bilinears, that to multiply any column, or any row, by m, multiplies V8 by m; and if a column or row consist of Binomials, we resolve the tablet into two. Conversely two V3s which differ only in a single row or single column can be joined into one universally, A + m, A'A A A ' A A + m A' A B +n, B' B BB B'Bo n B' B C+p, C C' C C' C p C' C 12. Further, if two contiguous columns (or rows) be identical, the VY =zero. For to exchange them changes V3 to - V. Yet the exchange leaves V8 exactly what it was before. Hence V,=-V3. This can only be when V. = 0. It follows that if a column (or row) be proportional to a contiguous column (or row), the tablet is zero. For instance, if a1, b, c1 are proportional to ab, b2, c, we may assume ai Ma = b =mb, c = mc then mOa mb. mc~ rna2 mb2 rnc2 V3= a2 b2 c2 a3 b3 C3 This gives a b2 c2 V3= m a2, b, c,, b, c3 44 TO SOLVE FOR x, y, Z IN SIMPLE EQUATIONS. and the last tablet is zero, because its first and second row are identical. What has been said in this Article of two contiguous rows or columns is evidently true of any two rows or any two columns, since exchange of contiguous rows (or columns) does but multiply the tablet by - 1. 13. The same argument as before in the Second Order now proves that a tablet V3 is not changed in value, if any row (or column) receives increase or decrease proportional to some other row (or column). Thus we have shown in V3 the same properties as those enunciated in V,. 14. NEW PROBLEM: to solve for x, y, z in the three equations {a1x + by + cz + d = 0, a2x + b2y + c2z + d = 0, [a3x + b3y + c z + d = O. Assume d = Ax, d =A x, d =A3x. Then (a + A,) x + b.y + c2z = 0 (a+ A) + by + c3z = 0 (a,s xA,)a)+by+c = ] Eliminate the x, y, z here visible; then a,+A, bl c1 a + A b2 C2 =0. a +AI b3 c3 The last tablet may be resolved into two, namely: a, b, c A, b c, a b2 C2 + A 2b2C2 = 0. a, b, c, As b, c, Multiply the former of these by x, and, as an equivalence, multiply each constituent of the first column of the latter by x, a, b1 c1 Ax b, c1 a. b2 c2 x+ A2x b c2 =0. a3 b3 c3 A2x b3 c3 THE PROBLEM SOLVED. 45 In the first column of the last restore for A1x, A2x, A3x their values d,, d2, d8. Then if a1 b, c1 d1 b1 c D = a2 b2 c2 you have Dx + d2 b2 C =0, a3 b3 c3 d3 b3 c3 which solves for x. By perfectly similar steps ad di ci ai b1 d, Dy + a2 d, c, = 0; Dz + a, b d2 = 0. a3 d2 C3 a3 b3 d3 These are easy to remember: each suggests the other, by entire syrmetry. The method succeeds in Higher Orders. COR. If we make =, =, z=-, (a,t + blq + c + dlu = 0, *.' a2: + b2 c + c2 + d2u = 0, a3.+ + b3q + c3' + d3u = 0. Then from -Dx- b6 c d3; Dy= a, cl d; Dz = a, b6 dl b2 c2 d a2 c d2 -,2 b d2 b3 c3 d3 as3 cd d a you obtain the proportion b, cl di a, cl di a, b6 di a b6 c b= c2d:- c2d: a2 b2 d2 a2 b2 c b3 C3 a3 c3 d3 a3 b3 d3 a3 b3 c3 15. PROBLEM. To eliminate x from the two equations Px 2+ Qx + R = 0 {px 4+ q + r = 0 where P, Q, R, p, q, r may involve y or other quantities. First, put 2 = X, then we have PX + Qx + r = O pX +qx +r 0ol Eliminate X; i.e. solve for x;. P Q x+ PR =0.................(1). pq p r 46 TWO PROBLEMS OF ELIMINATION. Again, writing in the original (Px+ Q) x + R = 0 (px + q) x +r =O eliminate x, as if Px + Q and px + q were ordinary coefficients. Then Px+Q, R = 0, px + q, r which expanded, gives PxIR + QR =0, or PR x+ QR =0...... (2). pxr q r p r q r Eliminate x between (1) and (2), which gives PQ PR pq pr PI? QI? -0; P R Q R Ip r q r that is, P Q. Q R - P R = 0, pq q r p r the result required. 16. PROBLEM. To eliminate x from the two equations, fax3 + 3bx2+ 3c + d = 0, ax2 + 2bx + c = 0. First, multiply the last by x and subtract;.. bx2 + 2cx + d = 0. Compare the two last with the two equations of Art. 15,.. P=a, Q=2b, R=c, p=b, q=2c, r=d. Hence the elimination of x yields a2b.2b c - ac 2=0, or ab. bc =~ ac 2 b 2c 2c d b d b c cd b d This is the condition of two equal roots in the given cubic. 17. By conducting elimination from three given simple equations by a second process, we attain relations which were not easy to anticipate. Assume the three equations of Art. 7. Eliminate y AN IDENTICAL EQ. BETWEEN 9 CONSTITUENTS, ALL ARBITRARY. 47 twice, (1) from the two first; (2) from the second and third. Hence we get a, b, x= b cl; a b2 x= b2 c; a2 b b c2 a b2 b c2 from which, by eliminating x, we obtain U = O, if we define U by the equation U= a1b. b c, - b1c,. a 2b a2b b3 c b b.c a b3 But, otherwise eliminated, we find, as the condition of compatibility, v=o. 30. Therefore U= O contains the same relation of the constituents as V = O. By inspection, we see that U, equally with V3, is linear in a,, bp, c1. Since then U and V vanish together, we have necessarily U-==/.V3, in which /, does not in volve al, b, nor c,. To determine /, suppose a, = 1, b = O, c = 0,.'. U a= a1 0. b.2 c2 =6b b2c2 a2 b2 b b3b b C3.and 3= = 1 0 0=1 00 =1. b2 c; a b2 c2 O b2 c2 b 3c as bs c 0 bs % 3 b 3 O b3 3 so that in this particular case U= b2V3, or, = b2. This then is the value of, for all values of al, b1, cl, or universally U= b2. V3, while ail the nine constituents are arbitrary. Q.E.D. 18. To remember this important equation, write the square trilinear larger and mark out its minor square. The factor b2 is in the centre. a, b1 c1.......... a2 b | c, for U=b2.T7. a lb3 c3 By interchanging rows or columns without altering the value of V3, fresh relations are obtained. Indeed no one constituent can claim the central place for itself. 48 QUADRILINEAR TABLET. Fourth Order. 19. To eliminate x, y, z, u from four given equations each of the form a1x + bly + clz + dlu = O, we have now much facility from the Cor. to Art. 14. First, eliminate from the three first equations and get the proportions of x, y, z, u. Next, insert these proportionate values in the fourth equation whereby you entirely eliminate all the four, and obtain an equation V4 = 0; if V4 stand for a4 b, c, di -b4 a1 c, d +c4 a, b1 dl -d4 a1 b, c. b2 c2 d2 a2 c2 d2 a2 b2 d2 a2 b2 c2 b2 c3 d a3 c2 d2 a3 b2 d3 a b2 c3 This developed form of V4 can always be recovered (by attention to the simple rule given for developing V3) from the conciser or undeveloped form a, b. c, dl a2 b2 c2 d2 V4= a3 b3 C3 d. a2 b, c, d, a4 b4 c4 d4 Evidently in the definition V4 is a linear function of a4 b4 c d4. So then it must be of any other row, the order of the given equations being arbitrary. Also the first term of V4 as defined is free from a1a2a.; each of the others is linear in aa2a3. Therefore V4 in square is linear as to its first column; so then must it be as to every column. Being thus linear, if one column (or row) of such quadrilinear tablet is binomial, it may be split into two tablets by the same process as in the third order. Likewise to multiply any column (or row) by m multiplies the whole tablet by m. To exchange first and second column, exchanges the first and second term of V4, but reverses their signs. It reverses the sign of the third term, also of the fourth. Thus to exchange the first and second column reverses the sign of V4. Evidently then the same must happen by exchanging any two contiguous columns. The same argument applies concerning any two contiguous rows. The reasoning of Art. 12 concerning TV now applies to VJ, showing that the tablet vanishes, if one column (or row) be pro ITS PROPERTIES THE SAME. 49 portional to another column (or row). From this it further follows (as concerning TV in Art. 13, and concerning V. in Art. 6), that V4 is not changed in value, if any row (or column) receive increase or decrease in proportion to some other row (or column). 20. That V4 is not altered by exchanging rows and columns, is generally proved by elaborate inspection of the separate terms when V4 is resolved into 24 elements each of the form ab2c.d4, no two factors of the same row or column, and showing that the + or - of the term is never altered. It is, no doubt, a perfect demonstration, and more elementary than mine; but I find the less obvious argument of Art. 9 the easier for all the higher orders. Multiply the given equations by m, nl~p q1 and assign to these multipliers the condition that from the sum of the equations thus multiplied y, z and u shall disappear. There will remain (ma1+ na2 +pa, + qa4) x =0. But our hypothesis forbids x=0, hence we have four equations to determine mnpq, viz. alm + a2n + a3p + a4q = 0, bm + b2n + b3p + b4q = 0, cdm + c2n + cn p + c4q = 0, dlm + d2n + dpp + d4q = 0. When we eliminate mnpq the result, which we may call S = 0, shows S4 differing from V4 only in the exchange of rows with columns. Each of them is linear in albctd1. Each involves the same relations between the constituents. The equation S4=0 must be deducible from V4=0. The only possible relation, making S4 and V4 vanish together, has the form 5= nV=, in which ~ is independent of aac^3a4. But symmetry proves /L equally independent of every other column; therefore a is numerical. To find it, we may make the constituents on the positive diagonal all = 1, and all the other constituents vanish. Then both S4 and V4 = alb2cd4 = 15. Universally then, / = 1, or S4= V,. Therefore V4 is not altered by exchanging rows with columns. Evidently this argument holds, however high the order of the Tablet, if the successive definitions follow the same law. N. 4 50 SOLUTION OF FOUR SIMPLE EQUATIONS. 21. We can now solve when four given simple equations connecting xyzu have on the left side absolute terms eeee, with zero (as before) on the right. We proceed as in Art. 14. Let e = A1x, e2 = A2x, e3 = Ax, e = A4. Then our equation becomes (a1 + A1) x + b1y + cz + d,u = O....................................... (a4 + A4) x + b4y + C4z + d4u = 0 Eliminate x, y, z, u, then a~ +A1 b, cl d a2+ A2 b2 c2 d2 =0. a3+A3 b3 c3 d3 a4 + A4 b4 c4 d4 The first column being binomial, we can resolve this tablet into two. Then multiply the left tablet by x, and the first column of the second also by x; whence a, b. c, d, Aax A b c1 d1 +............... = 0. Ax b, c, d, a4 b4 C4 d4 A In the second tablet we now replace its column by its value el e. e e,. Thus we have solved for x. By perfectly similar steps we solve for y, for z, and for u. Finally, if b1 c, d1 el a1 c1 dl el a1 b, dl el M=............ N =............,p =.......... b4 C4 d4 e4 a4 c4 d4 e4 a4 b4 d4 e4 Q =...........,............ a4 b4 c4 e4 a4 b4 c4 d4 we have x:y: z: u: = M:-N:: P: Q: R. COMPLEX RELATION U4 = (b2c3 - b3c2) V. 51 22. Take our four equations as in Art. 19. From the three first and also from the three last eliminate both y and z; whence a1 b1 c1 b, cl d,. 2 b2 c2 b2 c2 d2 a2 b c2 x=- b2 c2 d2; and as b c3 X=- b c3 d3 a3 b3 c3 b3 c3 d3 a4 b4 c4 b4 c4 d4 To eliminate x from these two, we have a1b C1 b2 c2 b2 2 c2 bl cdl U4=o=..-............, a3 b3 c4 4 b bc d 3 3 d which we may remember by i a, 1 C' d -. --- —... -----—.- -—.... -. a, b c d4 4 t} X 4 Thus U4= 0 and V = O express the same condition of the constituents for reconciling the four equations. Inspection shows that U4, like V4, is linear in albcld1. Put J4 = jV4, and pj will be free from these four. Assume then a,= 1, b = c = d= 0, and it will not affect ~. But it makes b2 c2 d2 b c b2c2d2 Y^^ T)b. cad V4=1. *;.. and U4=1. c2 I b*.'d4..; b4c 4d4 b33 b4 c4d4 that is, b2 c b3 c3 whence U4 generally = b2 c. V4 This could not have been forebs c3 seen. By varying the order of the elements, we have other results. 4-2 52 GENERALIZATION OF THESE PROPERTIES 5b C Observe that b2 c2 is the square in the centre of V4. The trib3 C3 linears in U are squares cut from the four corners of V4; those of the first term in U4 being from the positive diagonal. 23. By aid of Art. 21 we readily proceed to the Fifth Order, with the same law of continuous formation; whence in every Order we have these same properties. (1) To exchange rows and columns does not affect the value of V. (2) V, is a linear function of any one row, or any one column. (3) If a row or column be binomial, the Vl may be split into V + Vyn" (4) To multiply a row or column by m, multiplies Vn by m. (5) To exchange any row (or column) with a contiguous row (or column) changes Vn to -Vr (6) If one row (or column) is identical with or proportional to another row (or column), the Vn = zero. (7) V, is not altered in value when a row (or column) receives increase or decrease proportioned to another row or column. (8) If Vn be divided along the diagonal, so as to fall into four parts, two squares and two rectangular complements, the vanishing of one complement makes the other wholly ineffective. 24. To prove the last universally, it suffices to prove it for the fifth order. Call the two squares P, S and the complements Q, R. Then if one of the complements, as Q, have all its constituents zero, I say, R is ineffective, and V=P. S, just as if R also had all its constituents zero. For every term of V5 when fully expanded, has the form ab,,c,d,e,, where mnpqr are taken from 1, 2, 3, 4, 5 and no two are the same. Hence a4b4c4 and ab5c5 (the constituents of R) are necessarily multiplied by one or other of the zeros (d1ed2e2d3e3) in Q, and FOR ALL ORDERS. 53 all the products vanish. Consequently, if P is of the third order and S of the second, the V in question is equivalent to a, b c, 00 a, b2 c2 0 0 V,= a, b3 c, 0 0.................. 000 d4e4 o o 0 d, e, and might arise froni two equations separately, yielding P = 0, and S=O. In fact V= P.S. If Spottiswoode did not plant the first germ of this very valuable theory, he first investigated the laws and exhibited its vast power. TRACT V. INTRODUCTION TO TABLES I. AND II. To these four Elementary Tracts I have added two Numerical Tables, solely because their compilation and verification is elementary. Table I. gives values of A-" to 20 decimal places. Here A means the series 2, 3, 4,... up to 60, and the odd numbers from 61 to 77; and n means 1, 2, 3,... continued until A-" is about td vanish. To verify, use the formula A-l + A- 2+ A-3 +. A-m + A- 1 -The reader may convince himself how searching is this test, by applying it, for instance, to A-n when A = 37 or when A = 71. Only in the case of 2-" and 3-", where the Tablets give only odd values of n, we must apply the formula A A-2m-1 (A- +A-3 + A-6_+...+ A-2_-1) A "_1 Table II. has values of x" with 12 decimal places, where x means *02, "03, 04 up to '50 and n is continued from 1, 2, 3,... until x" is insignificant. The formula of verification is (with m any integer less than r) (oC+m+l ++2+... + r) ( -) = -f) +l I compiled this table while working at Spence's integral Jfog (1 + x), but it has much wider use. One who is sagely incredulous of printed tables can verify for himself any tablet which he is disposed to use with much greater ease than he could compose the tablet. Thus, too, he would detect any error from miscopying or misprinting, against which I can least give a guarantee. TABLE I. TWENTY DECIMALS. 55 n 2-" (n odd). n 3-" (n odd). I '5I 33333 33333 33333 33333 3 I25 3 '03703 70370 37037 03703 5 '0325 5 4II 52263 37448 55967 7 '0078I 25 7 45 72473 70827 61774 9 '0195 3I25 9 5 08052 63425 29086 II 48 82812 5 Il..... 56450 29269 47676 I3 I2 20703 125 I3...... 6272 25474 38630 I5 3 05175 78125 15...... 696 91719 37625 17...... 76293 9453I 25 I7 *... 77 43524 375I4 19...... I9073 48632 8125 19...... 8 6039I 59724 2I...... 4768 37I58 20312 21............ 95599 6636 23...... 1192 09289 55078 23........... 10622 II848 25...... 298 02322 38769 25........ I8o 23539 27..... 74 50580 59692 27............ I3I I3726 29...... 8 62645 I4923 29............ 14 5708I n 4-n, n 4- I 125 I7 '0000o 00000 58207 66091 2 -o625 8............ 4551 9I523 3 '01562 5 19............. 3637 97880 4 '00390 625 20............ 909 49470 5 97 65625............ 227 37367 6 24 4I406 25 22............ 56 84342 7 6 10351 5625 23............ 14 21085 8 I 52587 89062 5 24............ 3 5527I 9...... 38146 97265 625 25.............. 888i8 0...... 9536 743I 6 40625 26.................. 22204 II...... 2384 I8579 IOI56 27.................. 5551 12...... 596 04644 77539 28............... 1388 13...... 149 OII 6I 10385 29................ 347 14...... 37 25290 29846 30.................. 87 15...... 9 3I322 5746I 31............. 22 I6...... 2 32830 64365 32.............. 5 33.................. I 56 TABLE I. TWENTY DECIMALS. n 5_-' 6-" I '2 16666 66666 66666 66666 2 '04 2 02777 77777 77777 77777 3 'o8 3 462 96296 29629 62962 4 'oo6o 4 77 I6049 38271 60494 5 '00032 5 2 86008 23045 26749 6 oooo6 4 6 2 14334 70507 54458 7 ooooI 28 7..... 35722 45084 59076 8...... 256 8...... 5933 74180 76513 9...... 05120 9...... 992 29030 12752 IO...... 01024 IO-..... I65 38I7I 68792 II...... 00204 8 Il...... 27 5366I 97499 12.... 40 96 I2...... 4 59393 65799 I3..... 8 192 I3....... *... 76565 60966 14...... I 63840 14..... *..... 12760 93494 15............ 32768 5............ 2126 82249 I6............ 6553 6 I6......... 354 47041 17............ 1310 72 I7..... 59 07840 18............ 262 144 18............ 9 84640 19...... 52 42880 19............ 64106 20............ 1o 48576 20.................. 27351 21............ 2 097I5 2I.1.............. 4558 22.................. 41943 22.......... * - 759 23............ 8388 23................. 126 24.............. 677 24................ 21 25.................. 335 25.................. 3 26..6.......... 67 27.............. 13 28.................. 2 n 7 n 7-n I 'I4285 71428 57I42 8574 I2 ooooo00000 00000ooooo 72247 61581 2 '02040 81632 65306 12245 13............ 0321 08797 3 291 54518 95043 73178 I4............ 1474 44114 4 4I 64931 27863 39025 15......... 210 63445 5 5 94990 18266 I986 1 6............ 30 o9063 6...... 84998 59752 31409 17............ 4 29866 7...... 12142 65678 90201 I8................ 6I409 8...... 1734 66525 55743 19................ 8774 9...... 247 80932 22249 20.................. 1253 I...... 35 40133 I7464 2............... 79 11..... 5 05733 31o66 22.................. 25 23.................. 4 TABLE I. TWENTY DECIMALS. 57 n 8-n n 9-n I 'I25 I '11III IIIII IIIII IIIII 2 01562 5 2 'OI234 56790 I2345 6790I 3 o00I95 3T25 3 137 I742I 12482 85322 4 24 41406 25 4 15 24157 90275 87258 5 3 05I75 78125 5 I 69350 87808 43029 6......3846 97265 625 6...... I886 76423 I5892 7...... 4768 37158 20312 7.. 2090 75I58 12877 8... 596 04644 77539 8...... 232 30573 12542 9...... 74 50580 59692 9..... 25 81174 79171 10..... 9 31322 5746I I...... 2 86797 I9908 II.... I 16415 32182 II......... 3866 35545 I2........... 14551 9I523 I2............ 3540 70616 13............ I8 8 98940 13...... 393 4II79 4............ 227 37367 14.......... 43 71242 15........ 28 42171 I5............ 4 85693 T6............ 3 55271 I6 6.................. 53966 17.............. 44409 I7.......... 5996 J8............. 555I Is................ 666 I9................. 694 I9 9......7......- 74 20................ 87 20.................. 8 21................. II 21................... 0,9 22................. I n II-n n I '09090 90909 09090 90909 '08333 33333 33333 33333 2 826 44628 09917 35537 2 694 44444 44444 44444 3 75 13148 00901 57776 3 57 87037 03703 70370 4 6 83013 45536 50707 4 4 82253 0864I 97531 5...... 62092 13230 59153 5.... 40I87 75720 16461 6...... 5644 73930 ~5377 6..... 3348 97976 68038 7 * --- 5 3 158II 82307 7...... 279 08164 72336 8...... 46 65073 80209 8...... 23 25680 3936i 9...... 4 24097 61837 9...... I 93806 69946 IO 10....... 38554 32894 Io............ 16150 55829 II...... 3504 93899 II1...... I345 87985 12..3...... 38 63082 12............ 112 5665 13......... 28 96644 I3 *......... 9 34639 14..... 2 63331 I4............ 77886 15.....23939 5..6490 I15............ *23939 15.................. 6490 I66.................... 76...... 541 17 -............. I98 7............. 45 I8.....8........................ 4 19...... I 58 TABLE I, TWENTY DECIMALS. n I3-n n I4 I 'o7692 30769 23076 92307 I '07142 857I4 28571 42857 2 591 71597 63313 60947 2 5Io 20408 I6326 53061 3 45 5I66I 35639 50842 3 36 44314 86880 46647 4 3 50127 79664 57757 4 2 60308 20491 46189 5...... 26932 90743.42904 5...... 8593 44320 8870 6...... 207 762II 03300 6...... 1328 10308 62991 7.... I59 36631 61792 7...... 94 86450 61642 8...... 12 25894 73984 8...... 6 77603 61546 9............ 94299 59537 9.......... 48400 25825 1o............ 7253 853 o...... 345716130 I0... 7253 81503 10.. - 3457tI6I30 II............ 557 98577 I............ 246 94009 I2............ 42 92198 I2...... I7 63858 I3............ 3 30169 I3......... I 25989 I4.................. 25397 I4............... 8999 15.................. 594 I5................ 643 I6...... 50 I6.................. 46 I7............ II I7.................. 3 n8 1 15_"^~~5n II7 -I.o6666 66666 66666 66666 I '05882 35294 II764 70588 2 444 44444 44444 44444 2 346 02076 I2456 74740 3 29 62962 96296 29629 3 20 35416 24262 I6I6I 4 I 97530 864I9 75308 4 I 19730 3672I 30362 5...... 13168 72427 98360 5...... 7042 96277 72375 6...... 877 91495 I989I 6...... 44 29192 80728 7...... 58 52766 34659 7...... 24 370II 34160 8...... 3 90184 4231 8..... I 43353 60833 9............ 260I2 29487 9.......... 8432 56519 Io........... 1734 I5299 Io....... 496 03324 1............ II5 61020 II............ 29 17842 12......... 7 70734 I2...... I 71638 13.................. 51382 I3............... 10096 14................. 3425 I4................. 594 15................. 228 I5.............. 35 I6.................. 5.................. 2 17.......... 1 For I6-" look to 4-2" TABLE I. TWENTY DECIMALS. 59 n I8-~ n 19-" I 05555 55555 55555 55555 I *05263 I5789 47368 42I05 2 308 64197 53086 41975 2 277 00831 02493 07479 3 I7 14677 64060 35665 3 14 57938 47499 63551 4. 95259 86892 24204 4 76733 60394 7I766 5...... 5292 21494 01345 5 ** 4038 61073 40619 6...... 294 OII94 I86 6...... 212 55845 96874 7...... 6 33399 67288 7...... I8728 735I9 8............ 90744 42627 8............ 58880 45975 9.......... 504I 3570I 9 *... *- 3098 97156 10............ 280 07539 Io...... 1 63 Io376 I I............ 15 55974 I............ 8 5844I I2.............. 86443 12.................. 4518I 13............ 4802 13.................. 2378 14................ 267 14............... 125 I5............... 5. 15.................. 6 n 21- n 22-n I 04761 90476 I9047 61904 I '04545 45454 54545 45454 226 75736 96145 I2471 2 206 61157 02479 33884 3 Io 79796 998I6 43451 3 9 39I43 50II2 69722 4...... 54I8 90467 44926 4...... 42688 34096 03169 5...... 2448 5I927 02139 5 I.940 379I3 45599 6..... II6 59615 57245 6..... 88 19905 I5709 7... 5 55219 7896 7...... 4 00904 77987 8............ 26439 03758 8............ I8222 94454 9........... I259 00179 9........... 828 3I566 10............ 59 95246 Io............ 376507 IOl59 95246 37.6507I 11I............ 2 85488 I............ 7II 39 12.............. I3594 I2.............. 7779 I3.................. 647 I3....-............ 353 14 *.................. 31 14.................. I6 I 5.................. I 60 TABLE I. TWENTY DECIMALS. 23-n 24-n n 23 fl 24 I '04347 82608 69565 21739 I o0466 66666 66666 6666 2 189 0359I 6824I 96597 2 I73 6IIIi IIIII IIII1 3 8 21895 29053 99852 3 7 23379 62962 96296 4.... 35734 57784 95646 4..... 30I40 81790 I2345 5.... I553 67729 7807I 5...... I255 8674I 25514 6...... 67 55II8 6862 6...... 52 32780 88563 7...... 2 93700 81244 7...... 2 I8032 53690 8............ I2769 60054 8..... 9084 68903 9............ 555 20002 9............ 378 5287I 10............ 24 I3913 0.......... 15 77203 II............ I 04953 l............. 65716 I2............... 4563 2.................. 2738 13.................. 198 13.. I...4... 1,. 4 14................. 8 I4.............. 4 For 25-" look to 5~2". n 26-n n27.27 I 03846 I5384 61538 46153 I 03703 70370 37037 03703 2 147 92899 40828 40236 2 137 17421 12482 85322 3 5 68957 66954 93855 3 5 o8052 63425 29086 4...... 2882 98729 03609 4..... 886 76423 I5892 5...... 84I 65335 7326 5...... 696 91719 37625 6...... 32 37128 29739 6...... 25 81174 7917I 7..... I 24504 93451 7........... 95599 06636 8.......... 4788 65133 8........... 3540 70616 9............ I84 17890 9............ 13I I3726 Io........... 7 08380 o........... 4 85693 Il.................. 27245 II.................. 17982 12............. I048 12.............. 666 I3............ 40 1.............. 25 4....... I TABLE I. TWENTY DECIMALS. 61 n 28- n 29 -2 29 I 03571 42857 14285 71428 I '3448 27586 20689 65517 2 I27 55102 0408I 63265 2 ii8 90606 42092 74673 3 4 55539 35860 0583I 3 4 I0020 91I06 64644 4...... 16269 26280 71637 4...... I438 65210 5740I 5... 58I 04510 02558 5...... 487 53972 77841 6...... 20 7516I 07234 6...... 6 81171 47512 7............ 74II2 89544 7......... 5797I 43011 8.......... 2646 88912 8............ 999 01483 9............ 94 5375 9........ 68 9354 0o............ 3 376I3 0............ 2 37695 11.................. 2057 II.................. 8196 I2.............. 430 12.................. 283 3.................. 15 3.................. 9 n 31-~ n 32-n I '03225 80645 16129 03226 I '03I25 2 104 05827 26326 74298 2 97 65625 3 3 35671 84720 2I751 3 3 05175 78I25 4...... 10828 12410 32960 4.....9536 743I6 40625 5..... 349 29432 59127 5...... 298 02322 38769 6...... 11 26755 89004 6...... 9 3322 5746 7............ 36346 96419 7............ 29103 83045 8............ II72 48271 8......... 909 49470 9........... 37 82202 9....... 28 42171 0......... I 22007 10......... 888I8 11.................. 3936 II.......... - -..... 2775 12............ 1.. 27 I2............... 86 13...I3........... 2 62 TABLE I. TWENTY DECIMALS. n 33- n 34 -I '03030 30303 03030 30303 I 02941 I7647 05882 35294 2 9I 82736 45546 37282 2 86 50519 03II4 I8685 3 2 78264 74107 46584 3 2 54427 03032 77020 4...... 8432 26488 10502 4...... 7483 I4795 08147 5.... 255 52317 82136 5...... 220 09258 67888 6...... 7 743I2 66125 6...... 6 4733I 13761 7............ 23464 02004 7........ I9039 15II0 j8 8.... 7I o309............ 559 97503 9...... 2I 54639 9....... i6 46985 Io................ 65292 1............... 4844I II.................. I978 IlI.................. I425 I2... 59....... 2............ 42 3.............. 2 I 3............... I n | 35 n 37-" I '02857 14285 71428 57I42 I 02702 70270 27027 02702 2 8I 63265 30612 24489 2 73 0460I 89919 64938 3 2 33236 I5I60 34985 3 I 9742I 67295 12566 4...... 6663 89004 58142 4... 5335 72089 05745 5.... 1 I90 39685 84518 5...... 44 20867 27182 6...... 5 4399 02415 6...... 3 89753 16951 7......... 15542 60069 7............ I0533 86945 8.*........ 444 07430 8............ 284 69917 9........... 12 68783 9............ 7 69457 I0o.......... 3625............... 20796 I I.................. 1036 II.................. 562 I2.................. 29 12... I.............15 I3 2.... 2.......,8 For 36"- look to 6-2. TABLE I. TWENTY DECIMALS. 63 n | 38-"n 39" I '02631 57894 73684 21052 I '02564 I0256 41025 64102 2 69 25207 75623 26870 2 65 7462I 05923 73438 3 I 82242 30937 45444 3 I 68580 05023 68549 4...... 4795 85024 66985 4...... 4322 56539 o6886 5..... I26 20658 54394 5..... I 83501 ooI76 6...... 3 32122 59326 6...... 2 8492 33338 7............ 8740 06824 7............ 7286 98290 8....... 230 00179 8............. I86 8457I 9........... 6 05268 9............ 4 7909I I0................. 5928 Io0.................. 12284 II.................. 4 9 II.................. 3 15 12............................. 8 n 44I n 42" I '02439 02439 02439 02439 I '02380 95238 09523 80952 2 59 48839 97620 4640I 2 56 68934 24036 28II8 3 I 45093 65795 62I07 3 I 34974 62477 05431 4...... 3538 86970 62490 4... 3213 68I54 2I558 5...... 86 31389 52744 5...... 76 5I622 71942 6...... 2 I052I 69579 6...... I8218 49332 7............ 5I34 67550 7............ 4337 65460 8....... 125 23599 8............ 103 27749 9............ 3 05453 9............ 2 45899 0............... 7450 O.................. 5854 II................ 82 Il...............39 12.................. 4 I2................ 3 64 TABLE I. TWENTY DECIMALS. n 43 n 44-" I '02325 58139 53488 37209 I 02272 72727 27272 72727 2 54 08328 82639 26447 2 51 65289 256I9 8347I 3 I 25775 08898 58754 3 I I7392 93764 087I5 4...... 2925 00206 94389 4...... 2668 02131 00198 5...... 68 02330 39404 5.... 63 63684 79549 6...... 58193 73009 6...... I 37811 o8o8 7.......... 3678 92395............ 3132 6859 8........... 8 55637............ 7 18337 9............ 98968 9............ 61780 0.................. 4627 1o................ 3677 IrI....0 I........ I07 I.................. 84 12........2....12.................. 2 n 45- 4 46-" I '02222 22222 22222 22222 I '02173 91304 34782 60869 2 49 3827I 60493 82716 2 47 25897 92060 49149 3 I 09739 36899 86282 3 I 02736 91I31 74981 4...... 2438 65264 44I39 4...... 2233 4IIII 55977 5...... 54 19228 09869 5...... 48 55241 55565 6...... I 20427 29108 6...... I 05548 72947 7............ 2676 16202 7............ 2294 53759 8............ 59 47026 8............ 49 88125 9........ I 32156 9........... I 08438 10.................. 2937 Io.................. 2357 I I.................. 6 5.................. 5 I I2.................. I 12................ I TABLE I. TWENTY DECIMALS. 65 n 4.7-n n 48-" I '02127 65957 44680 85Io6 I '02083 33333 33333 33333 2 45 26935 26482 57130 2 43 40277 77777 77777 3...... 96317 7759 20364 3..... 90422 4537~ 37037 4...... 2049 3I428 91923 4...... I883 8o0II 8827I 5.... ~ 43 60243 I6849 5.... 39 24585 66422 6........... 92771 13124 6....... 81762 20136 7............ 1973 85385 7............. I703 379I9 8........... 41 99689 8............. 35 48706 9............ ' 89355 9.................. 73931 0.................. I90I I.................. I540 II........... ~ 40 II.................. 32 For 49-" look to 72n n 515-' n 52-n I 'oI960 7843.1 37254 90196i 'oI923 07692 30769 23076 2 38 44675 12495 I9415 2 36 98224 85207 I0059 3 *... 75385 78676 37636 3...... 7II9 70869 36732 4...... I478 15268 16424 4...... I367 68670 56475 5...... 28 98338 59143 5...... 26 3066 74163 6............ 56830 I6846 6........... 50580 I2965 7............ III4 I3703 7............ 972 69480 8......... 21 84935 8........... I8 70567 9............. 42842 9.................. 35972 o............840................. 692 II............ eI6 II.................. 3 N. 5 66 TABLE I. TWENTY DECIMALS. n 53-' n 54-f I oI886 79245 2830I 88679 I o-0851 85185 18518 5I852 2 35 59985 76005 69598 2 34 29355 28120 7133I 3..... 67169 54264 2584I 3 63506 57928 I6136 4.... 267 34986 II8o8 4. II76 04776 44743 5.... 23 91226 I53I7 5 * 21 77866 23051 6.. ---..... 45II7 47459 6 ****'" 40330 85612 7. *.***..**. 85I 27310 7.**, 746 86769 8....... 6 o06I76 8....... 13 83088 IO............... 5.......... 2563 474 9...39....25613 10...5............... 474 II................................ 9 X~n ~ 1 55 "n 56-" I oI8i8 I8888 8I818 I8I8I I 'OI785 7I428 57I42 857I4 2 33 05785 I2396 6942I 2 31 88775 51020 40816 3...... 6o 5 I8407 21262 3 **... 56942 41982 50729 4...... 1092 82152 85841 4..... OI6 82892 54477 5.... 19 86948 23379 5 - I8 I5765 93829 6............ 3626 3352 6...... 32424 39r75 7...... 656 84239 7....... 579 00699 8......... 1l 94259 8......... Io 33941 9.......... 2I7I4 9....... 74 I8463 10.................. 386 o.......... 330 Il...... 7 II........... 6 n 5,7 ' - n 58-" I I'o754 38596 49122 80701 I '01724 I3793 I0344 82758 2 30 77870 II388 II942 2 29 72651 60523 I8668 3..... 53997 72I29 61613 3 -' 5I252 61388 33082 4.......947 32844 37923 4...... 883 66575 66087 5..... 6 6I979 72595.... 5 2356I 64932 6............ 29157 53905 6.... 26268 30429 7........ 5II 53577 7............ 452 9or79...... 8 9743 8......... 7 80865 9............ 5744 9........ I3463 1o................ 276 o................. 232 IT 5 I......................... 4 TABLE I. TWENTY DECIMALS. 67 n 5 -59- n 6I I '01694 91525 42372 88136 '01639 34426 22950 81967 2 28 72737 71904 625II 2 26 87449 60I3I 98065 3... 48690 46981 43432 3...... 44056 55098 88493 4...... 825 26220 02431 4...... 722 23854 o8008 5...... 13 98749 49194 5...... I 83997 60787 6......... 23707 61851 6............ 19409 79685.7...... 401 82404........... 38 19339 8..... 8058........ 5 2628 9........ 11 543 9................. 855I I0O... 9............... I 40 11..3 I................. 2 'n 63-" n 65 - 1 01587 30158 730I5 87301 I '01538 46153 84615 3846I 2 25 I9526 32905 I1385 2 23 66863 90532 54438 3...... 39992 4814I 34942 3..... 36413 29085 II607 4...... 634 80129 22777 4...... 560 20447 46333 5...... I 07621 09885...... 8 61853 03789 6.......... I5993 98569 6.......... I3259 27751 7............ 253 87279 7............ 203 98888 8............ 4 02973............ 3 13829 9................... 6396 9.................. 4828 Io................I I.............. 75 n 67" n 69 I i01492 53731 34328 35821 I 'I449 27536 23188 40579 2 '. 22 27667 63198 93072 2 2I o00399 07582 44066 3... 33248 77062 6706o.:.. 30040 5663I 62957 4...... 496 25030 78613 4..... 441 16762 77724 5... 7 40672 I0128 5... 6 39373 37358 6............ II 54 80748............ 9266 28077 7............ I64 9972 7........... I34 29392 8...... 2 46264 8............ I 94629 9..36............ 3675 9................. 282I O.................. 55 IO................. 40 5? —2 68 TABLE L TWENTY DECIMALS. n 77- f 73-" I 01408 45070 42253 52213 I I01369 86301 36986 30137 2 19 83733 38623 28903 2 I8 76524 67629 94933 3...... 27939 90684 83506 3......25705 81748 35547 4...... 393 51981 47655 4...... 352 13448 60761 5...... 5 54253 26023 5...... 4 82376 00832 6............ 7806 38395 6............. 6607 89052 7............ 0Io9 94907 7........ 90 51905 8............ I 54858 8............ 23999 9 '...... 2.................. I699 10.................. 31 I.................. 23 nz 75 n 77-" I 'oI333 33333 33333 33333 'I01298 70129 87012 98701I 2 I7 77777777 77777 2 i6 86625 06324 84384 3...... 23703 70370 37037 3...... 21904 2216o 06291 4...... 316 04938 27160 4...... 284 4704i 03977 5...... 4 21399 17695 5...... 3 69442 09142 6............ 5618 65569 6...... 4797 94924 7............ 74 91541 7............ 62 31103 8.................. 99887.................. 80923 9.................. 1339 9................. 051 10o.................. i18 10............ 14 TABLE II. TWELVE DECIMALS. 69 TABLE II. Powers of '02, '03, '04,... up to '50, useful to compute Ax + A2x' + Ax3 + &c., when x does not exceed 9. (Twelve Decimals.) n ('02)" (03)~ ('o4)n n 2 '0004 0009 ooI6 2 3 ooo 08...... 27..... 64 3 4...... ooi6...... oo8 0256 4 5............ 32...... 2 43...... IO 24 5 6............ 0064............ 0729.......... 4096 6 7..oI............... 0022............ I64 7 8 6 8 8~~~~~~~~~~ ~oo............o 6 n ('05)" ('o6)" ('07)" 1 2 '0025 -0036 '0049 2 3 I 25 2 i6 3 43 3 4...... o62...... 296...... 240 4 5...... 3I 25...... 77 76...... 68 07 5 6...... I5625.... 4 6656...... I 7649 6 7............ 78............ 2799........... 8235 7 8............ 39........... i68............ 576 8 9....... 2............ IO............ 40 9 I............ 3 n ('o8)" ('09)" (.I)n n 2 '0064 'oo8 '0121 2 3 5 12 7 29 I3 31 3 4...... 4096...... 656I 464I 4 5...... 327 05...... 59~ 49....... 61o0 5 6 2...... 26 2144...... 53 I44I...... I77 156 6 7...... 2 0971..... 4 7830...... I9 4871 7 8............ I678.......... 4305..2... 2 I436 8 9......... 34........... 387............ 2358 9 IO........... II............ 259 io IO 10........ II *-l. -. -* — 35......... 259 IO 12............... 28 1i 12............ 3 12 70 TABLE II. TWELVE DECIMALS. n. ('I2) | ('I3)"( 14)"n 2 'o044 -o069 'oI96 2 3 I7 28 2I 97 27 44 3 4 2 0736 2 856I 3 84I6 4 5..2488 ' 32z...... 1 372 93..... 5378 24 5 6......' 298 598...... 482-6809...... 752 9536 6 7...... 35 838...... 627485...... I05 4I35 7 8...... 4.... 2988...... 8 1573.... I4 7579 8 9.......... 5I6........ i 0604...... 2 o66I 9... 6 9............ 1378............ 2892 IO II............ 74 79............ 079. 4 5 1 1 'I.. 74........179.... 405 It 12............ 8......... 23........... 57 I2 13........................ 3............ 8 1 3 14........... I I4 nl ('I5)n ('I6) ('7)" n 2 '0225 '0256 '0289 2 3 38 75 40 96 49 I3 3 4 5 0625 6 5538 8 3521 4 5..... 7593 75 I 0485 76 I 419857 5 6...... II39 0625...... 677 72I6...... 2413 7569 6 7 170 8594...... 268 4354...... 4IO 3387 7 8...... 25 6289...... 42 9497...... 697576 8 9...... 3 8443..... 6 8719...... 1 8588 9 10............ 576695...... 0995. 2 oi60 Io II.... 864............759........ 3427 II 12............ 130........... 28.I....583 12 I3,................ 9....... 45........... 99 I3 I4.............. 7....... i7 I4 15............ I ~......... 3 15 TABLE Il. TWELVE DECIMALS. 71 n ('18)" (19)> " (.20)" n 2 0324 o36 '04 2 3 58 32 68 59 008 3 4 IO 4976 13 032I qooi64 5 I 8895 68 24 4760 99 0003 2 5 6..... 340I2224...... 4704 588I.*. 64 6 7..... 6 2 2200...... 893 8717... 28 7 8...... IIo 996...... 69 8356..... 0256 8 9...... 198359 * * 32 2688.. oi 005 2 9 Q...... 3 5705..... 6 I3II.... 010 24 IO I........... 6427... I I649..... 0002 048 Il 12......* I5 7.............. 22I3.......... 4096 I2 2. x1157 2213. 4096 12 I3............ 208...................... 8I9 1 3, 13 208. 421. 8I9 13 14..... 37.. ***.. 80...... I64 I4 15... 7.............. 5.......... - 33 15 I6I 3.............. I6 - -.... I I6 n ('21) " ('22)" ('23)" n 2 '044I '0484 '0529 2 3 92 6I *oio8 48 *oi2I 67 3 4 19 448I 23 4236 27 9841 4 5 4 084I 01 5 I536 32 6 43633 5 6...... 8576 6I2I I 1337 9904 I 4803 5889 6 7...... I8o 0885..... 2494 3579...... 3404 8254 7 8...... 378 2286...... 548 7587...* 783 098 8 9...... 79 4280...... 79 4280...... I80 I53 9 o10...... i6 6799...... 26 5599...... 41 4265 1i 'II...... 3 5028...... 5 8432...... 9 528I I 2............ 7356...... I 2855..... 2 19I4 12 13............1545.. 2828...... 504~ 13 I4.........~~~.~~~~~ - ~ * *- 3245............. 622............. 5040 I 14............324........ 622............ 1159 I4 15............ 68...3......... 13. 267 I5 I6.....................30............ 6 16 17 3.................. 3 * 6,6............ 14 I7 1 8..........,6............ 3 I8 72 TABLE II. TWELVE DECIMALS. fl ('24)>" (225) ('26yn n 2 '0576 '0625 '0676 2 3 'I038 24 '0156 25 '0175 76 3 4 33 1776 '0039 0625 '0045 6976 4 5 7 9626 24 9 7656 25 'ooII 88I3 76 5 6 I 9110 2976 2 4414 0625 3 089I 5776 6 7...... 4586 4714...... 6103 5156..... 803I 8Io2 7 8...... IIoo 753I...... 1525 8789...... 2088 2706 8 9...... 264 1807... 381 4697...... 542 9504 9 o...... 63 4034...... 95 3674..... 14I 671 1o 1I...... 15 2i68...... 23 8418...... 36 7034 II 12...... 3 6520...... 5 9604 9 5429 12 3............ 8765...... I 490I...... 2 4811 I3 14............ 2103........ 3725............ 6451 14 15............ 505...... 93............ 677 15 6............ I2I............ 233 436 i6 I7........... 29............ 58........... 3 17 18...... 7.......... ':..... 14...... 29 I8 19............ 7...... 4............ 7 19 20............ I............ 2 20 n ('27)" (-28) ('29)n n 2 '0729 '0784 '0841 2 3 '0196 83 '0219 52 '0243 89 3 4 '0053 I441 'oo6I 4656 '0070 7281 4 5 '0014 3489 07 'ooi7 2103 68 '0020 5III 49 5 6 3 8742 0489 4 8189 0304 5 9482 332I 6 '7 I 0460 3532 I 3492 9285 I 7249 8763 7 8...... 2824 2954...... 3778 0200oo...... 5002-4641 8 9...... 762'5597...... 1057 8456...... 1450 7146 9 Io...... 205 8911...... 296 1968...... 420 7072 IO Il...... 55 5906...... 82 935...... 122 0051 Il 2...... 15 0095...... 23 2218 35 3817 12 13..... 4 0525...... 6 5021....... o 2607 13 14...... I 0942...... 1 8026..... 2 9756 14 15............ 2954. 5098...... 8629 15 16............ 797............ 427........... 2502 I6 17............ 215............ 400......... 726 17 i8............ 58...... i2....... 2...... 2I I8 19............ i6......3............ 6i 19 20...... 5............ 9............ 8 20 21............ I........... 2........... 5 21 22............ I 22 TABLE II. TWELVE DECIMALS. 73 n ( 30)" (.3I)> ('32)n 2 09 '96I I0242 3 027 1 207 91 '0327 68 3 4 00oo8I 0092 352I '0104 3576 4 5 0024 3 '0028 629I 51 33 5544 32 5 6 7 29 8 8750 368I 10 7374 I824 6 7 2 I870 2 7512 614I 3 4359.7384 7 8...... 656I...... 8528 9104 I 0995 II63 8 9...... I968 3...... 2643 9622...... 35I8 4372 9 10...... 0590 49... 819 6283.. II25 8999 Io II...... 0177 I47...... 2-54 0848. 360 2880 II 12...... 53 I44I..... 78 7663.... II5 292I 2 I3...... 15 9432.. 24 4175...... 36 8935 I3 I4...... 47830.... 7 5694.. II 8059 I4 15...... 1 4349..... 2 3465.... 3 7779 I5 I6............ 43~4........... 7274...... 2089 I6 16............'4304 7i79 17............ I291........ 2255........... 3868 I7 I8............ 387....... 699.......... I238 I8 g19......... I6............ 217.......... 396 I9 20.......... 35........... 67........... 127 20 2I............. IO............ 21...... 40 2I 22..................6..... 63 22 23........... 2....... 4 23 24 1........ 24 74 TABLE Il. TWELVE DECIMALS. ('33)" ('34)" (.35 )" 2 o1089 'II56 'I225 2 3 '359 37 '0393 04 '0428 75 3 4 'oII8 5927 'OI333336 'o50 0625 4 5 39 I353 93 45 4334 24 52 52I8 75 5 6 12 9146 7969 I5 4480 4416 I8 3826 5625 6 7 4 2618 4430 5 2523 350I 6 4339 2969 7 8 I 4064 0862 I 7857 9390 2 2518 7539 8 9..... 4641 I484...... 6071 6993...... 788I15639 9 10...... 153 579...... 2064 3777...... 2758 5473 1 TI...... 505 4210...... 7oi 8884...... 965 49I6 I 12...... i66 7889...... 238 642I...... 337 9220 I2 3...... 55 0403...... 8 I383...... I 8 2727 I3 4...... i8 I633...... 27 5870...... 4I 3954 I4 5...... 5 9939...... 9 3796...... 4 4884 I5 I6...... I 9780...,. 3 890...... 5 0709 I6 17............ 6527......, I 0843...... 7748 I7 I8............ 2154............ 3686...... 62I2 I8 19............ 7............ 253............ 2I74 I9 20........... 234............ 426............ 76 20 2............ 77............ I45............ 266 2I 22............ 25........... 49.......... 93 22 23.8......... 8........................... 3323 24............ 3.... 6............. I. 24 25........................ 2............ 4 25 26........... I 26 TABLE II. TWELVE DECIMALS. n (36) (.37)" (38)" 2 '1296 'I369 '4442 3 '0466 56 'o506 53 '0548 72 3 4 'oi67 9616 'oI87 4161 '0208 5I36 4 5 60 466I 76 69 3439 57 79 235I 68 5 6 2i 7678 2336 25 6572 6409 30 1o93 6384 6 7 7 8364 I64I 9 493I 877I 4I 4415 5826 7 8 2 82II 0991 3 5I24 7945 4 3477 9314 8 9 I oi55 9957 I 2996 1740 I 6521 6IOI 9 0...... 3656 1584...... 4808 5844...... 6278 2118 10 3......13 6 31 70...... I779 1762...... 2385 7205 Il 12...... 473 838I......658 2952...... 906 5738 I2 I3,...... I70 5817...... 243 5692...... 344 4980 13 14...... 6I 4094......90 206...... 30 9092 14 15..... 22 I074... 33 3446. 49 7455 15 I6.. 7 9587.... I2 3375.... 8 933 I6 17...... 2 865I..... 45649... 7 I833 T7 18...... I 0314... I 6890...... 2 7296 I8 I9............ 3713............ 6249..... I 0373 19 20............ I337............ 2312............ 3942 20 21........ 48I............ 855........... 1498 21 22 3............... I73 36...... 569 22 23........ 62........ I7............ 26 23 24............. 22............ 43............ 82 24 25............ i6.............25 26....................... 6......2 26 27......I...... I............ 2 4 27 28...........,7 28 76 TABLE II. TWELVE DECIMALS. n ('39) (40)" ('41)" n 2 *I52I *I6 '168I 2 3 '0593 I9 'o64 0689 2I 3 4 '0231 3441 '0256 '0282 576I 4 5 90 2241 99 '012 4 'oII5 8562 or 5 6 35 I874 376I '0040 96 47 5010 424I 6 7 I3 7231 0067 'oo16 384 19 4754 2739 7 8 5 3520 0926 6 5536 7 9849 2523 8 9 2 0872 836I 2 6214 4 3 2738 1934 9 I...... 8140 406I I 0485 76 I 3422 6593 1I I...... 3174 7584...... 494 304......5503 2903 I 12.... 1238 I558...... I677 7216...... 2256 3490 I2 13...... 482 8807...... 67I o886..... 925 I03I 13 14...... 88 3235...... 268 4354...... 379 2923 I4 I5.... 73 4462...... 107 3742...... 155 5098 15 6...... 28 6440...... 42 9497...... 63 7590 I6 17...... I 1717 2...... 17 1799..... 26 1412 17 8...... 4 3567...... 6 8719...... 7I79 I8 9..... I 699I...... 27487...... 4 3943 I9 20............ 6626...... I 0995...... I 8017 20 21......... 2584............ 4398........... 7387 21 22.......... 008 759..... 3029 22 23...... 393....... 71.....3 242 23 24............. 153............ 28I......... 509 24 25............ 60............. II2. 209 25 26............ 23............ 45...... 85 26 27...... 9....... 8...... 35 27 28...... 7 - 3........... 7........... 14 28 29................ 3........... 6 29 30........................ 2,4 30 TABLE II. TWELVE DECIMALS. 77 n (.42)4 ('43) ('44) f 2 I1764 '*849 I936 2 3 '0740 88 '0795 07 'o851 84 3 4 'o311 1696 'o341 880I '0374 8096 4 5 'oI30 6912 32 '0174 0084 43 'I164 9162 24 5 6 54 8903 1744 63 2136 3049 72 5631 3856 6 7 23 0539 3332 27 I8i8 6I11 31 9277 8097 7 8 9 6826 5200 I 6882 0028 14 0482 2362 8 9 4 o667 I384 5 0259 2612 6 I8I2 I839 9 I0 I 7080 1981 2 i6II 4823 2 7197 3609 Io II...... 7173 6832...... 9292 9374 I 1966 8388 il I2...... 3012 9469......3905 963I...... 5265 4091 12 I3...... 1265 4377...... I718 2641...... 236 7800 I3 14...... 531 4838...... 738 8536...... 1019 3832 I4 15...... 223 2232...... 317 7070...... 448 5286 15 6...... 93 7537..... I36 6140...... 197 3526 16 17.....30 3766...... 58 7440...... 86 8351 I7 8...... 16 5382...... 25 2599..... 38 2075 18 19...... 6 9460...... IO 868...... I6 8II3 19 20 29173...... 93.. 4 6706..... 7 3970 20 21...... I2253...... 2 0083...... 3 2547 21 22............ 5146........... 8636..... I 4321 22 23............ 2I6............ 3713........... 630I 23 24............ 908............ 1597........... 2772 24 25............ 381............ 687............ 1220 25 26............ 6o.............. 295............ 537 26 27............ 67.......... 127............ 236 27 28............ 28............ 54........... 04 28 29......... 2......... 46 29 30............ 5........ I......... 20 30 31.............. 2......... 4.*.... 9 31 32...... 2............ 4 32 33 1..I.. 33 TABLE II. TWELVE DECIMALS.. n ('45)' ('46) ('47) - n 2 '2025 '2116 2209 2 3 '0911 25 '0973 36 '038 233 4 '0410 0625;'0447 7456 '0487 9681 4 5 '0184 5281 2500 '0205 9629 76 '0229 3450 07 5 6 8i3 0376 5625 94 7429 6896 '0107 7921 5329 6 7 37. 3669 453I 43 5817 6572 50 6623 I205 7 8.I6 8151 25,39 20,0476 1223 23 8Ii2 8666 8 9 7 5668 0642 9 2219 0163 IlI 1913 0473 9 10 43 4050 6289 4 2420 7475 5 2599 1322 IO I,, I 5322 7830 1 9513 5438 2 472I 592I II 12.,.... 6895 2523..... 8976 2302 I 16I9 1483 I2 13....... 3102 8636...... 4129 0689...5460 9997 13 14...... 1.396 2886...... 809 3703.....2566 6699 4 15..<.. 628 3299...... 873 7I03....1206 2348 15 16...... 282 7484..... 40I 9067.... 66 9774 6,17'I7.......27 2368..... 184 877I i..... 2664794 I7 18..... 57 2566...... 85 0435... I25 2453 8 8 -19......... 25 7654...... 39 200...... 58 8653 1 20..... II 5944...... 17 9952...... 27 6667 20 21.......5 2175........ 8 2778... 13 0033 2.-2 2 3479..... 3 8078.... 6 I6 22.23 -...... I o565,....... I 75'16..... 2 8724 23.24........ 4754........... 8057...... I 3500 24 2-5..........-.... 2I39...... 3706...........6345 25 I 26............ 963........... I75.......... 2982 26!:27.........:" 433.............. 784............ 402 27 28............ I3.............. 3.659 28:30 39 76.....1 1~ 45.30.231 2 88.................... 31O 6 29 30.............. 3.9......... 76............. 145 I 3~;....I,31 8............ I827.... 35........ 68 3 3'2.................... 8........ i6 32 32 33.... 3,67........ o.....7....... 15 33 '34 ~,......................... 34 1..... i,6............ 3 7 34 35............ i,6....... 3 2 35 36............,6 36 TABLE II. TWELVE DECIMALS. 79 n (.48) (.49) ('50)n n 2 2304 '2401 '25 2 3 '1105 92 'II76 49 'I25 3 4 '0530 8416 '0576 4801 *o625 4 5 '0254 8039 68 '0282 4752 49 '03I2 5 5 6 '0122 3059 0464 'OI38 4128 7201I 'I56 25 6 7 58 7068 3423 67 8223 0728 '0078 I25 7 8 28 I792 8043 33 2329 3057 39 0625 8 9 13 5260 5460 I6 2841 3598 I9 5312 5 9 10 6 4925 o62I 7 9792 2663 9 7656 25 1o 11 3 II64 o298 3 9098 2105 4 8828 I25 II 12 I 4958 7343 I 9I58 I23I 2 4414 0625 I2 3...... 780 1925...... 9378 4803 I 2207 0312 13 14...... 3446 4924...... 4599 8653...... 6103 5I56 14 15...... 654 364...... 2253 9340.... 3051 7578 15 I6...... 794 079...... II04 4277 525 8789 I6 17......38 I545 54 I696......762 9394 I7 I8...... 82 954I...... 263 1731...... 38I 4697 I8 I9...... 87 8i8o...... I29 9348..... 190 7348 I9 20...... 42 I526...... 63 668I..... 95 3674 20 21...... 20 2333...... 31 1975...... 47 6837 21 22...... 97120...... 5 2867...... 23 84I8 22 23...... 4 668...... 7 4905...... 1I 9209 23 24.... 2 2376...... 3 6703...... 5 9604 24 25...... I 074...... I 7985... 2 9802 25 26............ 5I5 5............ 88I3...... I 490I 26 27........ 2474..8........ 4318............ 7450 27 28........ 90............ 2ii6............ 3725 28 29.......... 57............ I037........ I862 29 30......... * 274......... 508............ 931 30 31............ 1 32............ 249.......... 465 3I 32............ 63............ 122............ 233 32 33........... 30............ 60............ 16 33 34............ 14....... 29.......... 58 34 35 7...7.. 7...........* I4............ 29 35 36...... 3........... * 7............ I4 36 37................... 3.......... 7 37 38............,7............ 3 38 39........... I,8 39