1 II I ill I! ' 1^1 . hllM! Ill 1 III I nifiii :|i 111 liiilll III MATHEMATICAL MONOGRAPHS. EDITED BY Mansfield Merriman and Robert S. Woodward. Octavo, Cloth, $1.00 each. No. 1. HISTORY OF MODERN MATHEMATICS. By David Eugene Smith. No. 2. SYNTHETIC PROJECTIVE GEOMETRY. By George Uruce Halsted. No. 3. DETERMINANTS. By Laenas Giffokd Weld. No. 4. HYPERBOLIC FUNCTIONS. By James McMahon. No. 5. HARMONIC FUNCTIONS. By William E. Byerly. No. 6. QR\SSMANN'S SPACE ANALYSIS. By Edward W. Hyde. No. 7. PROBABILITY AND THEORY OF ERRORS. By Robert S. Woodward. No. 8. VECTOR ANALYSIS AND QUATERNIONS. By Alexander Macfarlanr. No. 9 DIFFERENTIAL EQUATIONS. By Willia:\i Woolsey Johnson. No. 10 THE SOLUTION OF EQUATIONS. By Mansfield Mekrim.\n. No. 11. FUNCTIONS OF A COMPLEX VARIABLE. By Thomas S. Fiskb. PUr.T.ISHED BY JOHN WILEY & SONS, NEW YORK. CHAPMAN & HALL, Limited, LONDON. MATHEMATICAL MONOGRAPHS. EDITED BY MANSFIELD MERRIMAN and ROBERT S. WOODWARD. No. 6. GRASSMANN'S SPACE ANALYSIS BY EDWARD W. HYDE, ACTl'AKY OF THE COLUMBIA INSURANCE COMPANY. FOURTH EDITION. FIRST THOUSAND. NEW YORK: JOHN WILEY & SONS. London: CHAPMAN & HALL, Limited. 1906. Copyright, 1896, BY MANSFIELD MERRIMAN and ROBERT S. WOODWARD UNDER THE TITLE HIGHER MATHEMATICS. First Edition, September, 1896. Second Edition, January, 1898. Third Edition, August, 1900. Fourth Edition, January, 1906. ROBKRT DRUMMOND, PRINTER, NEW YORK. EDITORS' PREFACE. The volume called Higher Mathematics, the first edition of which was published in 1896, contained eleven chapters by eleven authors, each chapter being independent of the others, but all supposing the reader to have at least a mathematical training equivalent to that given in classical and engineering colleges. The publication of that volume is now discontinued and the chapters are issued in separate form. In these reissues it will generally be found that the monographs are enlarged by additional articles or appendices which either amplify the former presentation or record recent advances. This plan of publication has been arranged in order to meet the demand of teachers and the convenience of classes, but it is also thought that it may prove advantageous to readers in special lines of mathematical literature. It is the intention of the publishers and editors to add other monographs to the series from time to time, if the call for the same seems to warrant it. Among the topics which are under consideration are those of elliptic functions, the theoiy of num. bers, the group theory, the calculus of variations, and non- Euclidean geometry; possibly also monographs on branches of astronomy, mechanics, and mathematical physics may be included. It is the hope of the editors that this form of pubHcation may tend to promote mathematical study and research over a wider field than that which the former volume has occupied. December, 1905. 236367 AUTHOR'S PREPACK. This little book is an attempt to present simply and con- cisely the elemental-}' principles of the "Extensive Analysis'" as fully developed in the comprehensive treatises of Hermann Grassmann, restricting the treatment however to the geometry of two and three dimensional space. It is designed to set forth, as far as is possible in so brief a work, the remarkable adapta- bility and effectiveness of the methods used as applied to the various problems and operations of geometry and mechanics. The ideas of direction and position appear to the writer to be as simple and fundamental as that of magnitude, and an algebra which deals directly with all three of these ideas should not be greatly more difficult than the ordinary one, which deals with magnitude only. The result of using such an algebra is an extraordinary gain in the brevity of operations and the expres- siveness of formulas and equations. Some of the terms belonging to this general subject are fre- quently employed in modern text-books on mechanics and physics, even when no use is made of the algebraic systems from which they are derived. It is hoped that this little book may do something to interest students, and to help toward bringing in the time when the methods as well as the ideas of this calculus shall come into general use. Cincinnati, O., December, 1905. CONTENTS. Art. I. Explanations and Definitions Page 8 2. Sum and Difference of Two Points 9 3. Sum of Two Weighted Points 12 4. Sum of any Number of Points 15 5. Reference Systems 20 6. Nature of Goemetric Multiplication 24 7. Planimetric Products 26 8. The Complement 33 9. Equations of Condition and Formulas 39 10. Stereometric Products 44 11. The Complement in Solid Space 50 12. Addition of Sects in Solid Space 53 GRASSMANN'S SPACE ANALYSIS. Introduction. The title adopted for this brief and elementary discussion of Grassmann's methods indicates at once its Hmitations; for his theory in its fullness treats of the linear relation p = lxkek, (k = o, 1, 2, . . .fi), in which n may be any positive whole number, Xq, Xi . . . any numbers whatever, and Cq, ei . . . units of any kind which are susceptible of being related to each other by such an equation as the above. Our treatment extends only to the case when n does not exceed three, and Cq, e^ . . . are points, or point products. Grassmann's tirst pubHcation of his new system was made in 1844 in a book entitled "Die Lineale Ausdehnungslehre Ein Neuer Zweig der Mathematik." His novel and fruitful ideas were however presented in a somewhat abstruse and unusual form, with the result, as the author himself states in the preface to the second edition issued in 1878, that scarcely any notice was taken of the book by Mathematicians. He was finally comdnced that it would be necessary to treat the subject in an entirely dift'erent manner in order to gain the attention of the mathematical world. Accordingly in 1862 he pubhshed "Die Ausdehnungslehre, vollstandig und in strenger Form bearbeitet," in which the treatment is algebraic, and is developed from the equation given above. Since that time his great work has been more fully appreciated, but not even yet, in the opinion of the writer, at its real value.. :S. crassmann's space analysis. Hamilton first gained the ear of the English-speaking world for his Quaternion methods, and was fortunate in having some very zealous adherents and interpreters who made propaganda for them, and were inclined to undervalue work not originating in England. It is hoped that the following brief presentation of Grass- mann's Analysis will serve to interest some to the extent that they may be led to investigate his original works. The writer has followed in tlie main the notation of Grass- mann himself, the principal deviations being the omission of the brackets from geometric products, writing pq instead of [pq]^ and a somewhat different treatment of the product p\q. Art. 1. Explanations and Definitions. 'The algebra with which the student is already familiar deals directly with only one quaHty of the various geometric and mechanical entities, such as lines, forces, etc., namely, with their magnitude. Such questions as How much? How far? How long? are answered by an algebraic operation or series of operations. Questions of direction and position are dealt with indirectly by means of systems of coordinates of various kinds. In this chapter an algebra * will be developed which deals directly with the three qualities of geometric and mechanical, quantities, viz., magnitude, position, and direction. A geomet- ric quantity may possess one, two, or all three of these prop- erties simultaneously; thus a straight line of given length has all three, while a point has only one. The geometric quantities with which we are to be concerned are the point, the straight line, the plane, the vector, and the plane-vector. When the word "line" is used by itself, a "straight line" will be alwaj^s intended. A portion of a given straight line of definite length will be called a " sect " ; though when the length * The algebra of this chapter is a particular case of the very general and comprehensive theory developed by Hermann Grassmann, and published by him in 1844 under the title "Die lineale Ausdehnungslehre, ein neuer Zweig der Mathematik." He publishe 1 also a second treatise on the subject in 1S62. EXPLANATIONS AND DEFINITIONS. of the sect is a matter of indifference, the word line will fre- quently be used instead. Similarly, a definite area of a given plane will be called a " plane-sect." If a point recede to infinity, it has no longer any significance as regards position, but still indicates a direction, since all lines passing through finite points, and also through this point at infinity, are parallel. Similarly, a line wholly at infinity fixes a plane direction, that is, all planes passing through finite points, and also through this line at infinity, are parallel. Thus a point and line at infinity are respectively equivalent to a line direction and a plane direction. A quantity possessing magnitude only will be termed a "scalar " quantity. Such are the ordinary subjects of algebraic analysis, a, x, sin 6, log^, etc., and they may evidently be in- trinsically either positive or negative. The letter T prefixed to a letter denoting some geometric quantity will be used to designate its absolute or numerical magnitude, always positive. Thus, if Z be a sect, and Pa plane- sect, then TL is the length of Z, and TP is the area of P. That portion of a geometric quantity whose magnitude is unity will be called its "unit," and will be indicated by prefixing the letter U\ thus UL = unit of Z = sect one unit long on line Z.* Hence we have TL . UL = L. Art. 2. Sum and Difference of Two Points. In geometric addition and subtraction we shall use the or- dinary symbols +, — , =, but with modified significance, as will appear in the development of the subject. Every mathematical, or other, theory rests on certain fun- damental assumptions, the justification for these assumptions * The word "scalar" and the use of the letters T and U, as above, were introduced by Hamilton in his Quaternions, ^stands for tensor, i.e., stretcher, and TL is the factor that stretches UL into L. The notation [ L \ for absolute magnitude is not used, because the sign | has been appropriated by Grassmann lo another use. 10 grassmann's space analysis. lying in the harmony and reasonableness of the resulting theory, and its accordance with the ascertained facts of nature. Our first assumption, then, will be that the associative and commutative laws hold for geometric addition and subtrac- tion, that is, whatever A, B, C may represent, we have A-\-B-{-C^{A-^B)-\-C=A-^{B-\-C) = A-{-C+B = {A-\-C) + B. We shall also assume that we always have A — A = o, and that the same quantity may be added to or subtracted from both sides of an equation without affecting the equality. Now let/, ,/„ be two points, and consider the equation A +A - A = A + (A - A) = A- (i> In this form we have an identity. Write it, however, in the form A -A + A = (A - A) + A = A > (2) and it appears that/^ — /, is an operator that changes/, into p„hy being added to it. Conceive this change of/, into/„ to take place along the straight line through /, and />„ ; then the operation is that of moving a point through a definite length or distance in a definite direction, namely, from/, to /„. This operator has been called by Hamilton "a vector,"'^ that is, a carrier, because it carries/, rectilinearly to/^. Grassmann gives to it the name Strecke, and some writers now use the word " stroke " in the same sense. Again, /^—/j is the difference of two points, and the only difference that can exist between them is that of position, i.e. a certain distance in a certain direction. Hence we may regard/, — /, as a directed length, and also' as the operator which moves /, over this length in this direc- tion. Writing /j — /, = e, equation (2) becomes A + e=A- (3) * See the first of Hamilton's Lectures on Quaternions, where a very full discussion of equation (2) will be found. Also Grassmann (1S62), Art. 227. SUM AND DIFFERENCE OF TWO POINTS. 11 Thus the sum of a point and a vector is a point distant from the first by the length of the vector and in its direction. Since /, — /, — — (/, —/J, it appears that the negative of a vector is a vector of the same length in the opposite direction. If p^ — px =^0, or p^ =p^,p^ must coincide with /, because there is now no difference between the two points. The question arises as to what, if any, effect the operator A~/i should have on any other point/3, that is, what is the value of the expression /^ — /i+A^ We will assume that it is some point /^, so that we have A— A+A=A. oi' A-/. =A-A- (4) This implies that the transference from /g to/, is the same in amount and direction as that from/, to /j, that is, that /, , A' A' A ^^^ ^^^^ four corners of a parallelogram taken in order. Thus equal vectors have the same ^^^^^ length and direction, and, conversely, vectors having the same length and direction are equal. Note that parallel vectors of equal length are not neces- sarily equal, for their directions may be opposite. Equation (4) may also be written A+A=A+A. (5) so that, whatever meaning may be assigned to the sum of two points, if we are to be consistent with assumptions already made, we must have the sum of either pair of opposite corner- points of a parallelogram equal to the sum of the other pair. The sum cannot therefore depend on the actual distances apart of the points forming the pairs, for the ratio of these two distances may be made as large or as small as we please. If n be a scalar quantity, we will denote that the operation e is to be performed n times on a point to which we is added, that is, the point will be moved n times the length of e; hence 1% grassmann's space analysis. ne is a vector u times as long as e, and having the same or the opposite direction according to the sign of ;/. In the figure above, let A— A = e,, A— A = e2. A-/'i = e3, A— A = e4- Then e> + e, =A— A+A —A =A-A+A— A=A— A = ^3, (5) since, by Eq. 4, p, -/, =/^ - /^. Also, e, - e^ ^A - A = e,. (6) Hence, if two vectors are drawn outwards from a point, and the parallelogram of which these are two adjacent sides is com- pleted, then the two diagonals of this parallelogram will repre- sent respectively the sum and difference of the two vectors, the sum being that diagonal which passes through the origin of the two vectors, and the difference that which passes through their extremities.* Again,/, -/, +/3 -A +/, -/3 = o = e, + e, + (- ej ; hence the sum of three vectors represented by the sides of a triangle taken around in order is zero. Similarly, if/,, A' • • -A t>e any ;/ points whatever taken as corners of a closed polygon, we shall have (A-A)+(A-A)+(A-A)+---+(A-A-:)+(A-A)=o; that is, the sum of vectors represented by the sides taken in order about the polygon is zero. By "taken in order" is not meant that any particular order of the points must be obserx-ed in forming the polygon, which is evidently unnecessary, but simply that, when tlie polygon is formed, the vectors will be- the operators that will move a point from the starting position along the successive sides back to this position again, so that the final distance from the starting-point will be nothing. Art. 3. Sum of Two Weighted Points. f Consider the sum w,/',-|-;;/„/2, in which m^ and w„ are scalars, that is, numbers, positive or negative, and /,, /^ are points. * Grassmann (1S44), § 15. f Grassmann (1844), ^ 95, and (1862), Art. 227. SUM OF TWO WEIGHTED POINTS. The scala,-s ». and «. will be regarded as values or weigMs assigned to the points/, and A- When any we.ght s o un value the figure x will be omitted, so that / -- ^ ^^ ;^^^' called a unit point. Occasionally, however, a letter may used to denote a point whose weight is not un.ty. To assist his thinking, the reader may consider the weights initially as like or unlike parallel forces acting at the pomts, m order to arrive at a meaning for the above expressron .,e LI make two reasonable assumptions, wh.ch w.ll prove to be ct Lrent with those already made, viz., first, that the sum .s a poTnt, and second, that its weight is the sum of the weights of the two given points. Denoting this sum-point by/, we *' '"" «/,/, + m.p, = ('«, + j".)A ^^'^ Transposing, we have »,(/-, -/) = "'.(/ - A). <>" Both members of (8) are vectors, and. being equal, they must, by Art. 4, be parallel. This requires that / shall be colhnear wIthA and A. Also, since/.-/ and/ -A are v_ectors whose lengths are respectively the distances from A to/ and from/ to fit follows that these distances are in the rat.o of ,n, to ».. Hence 7 is a point on the line /.A whose distances from/. and A -e inversely proportional to the weights of these pom s. We shall call 7 the mean point of the two weighted po.nts. „ „, and nu are both positive, (8) shows that /must he e- tweei, A andA-. but if one, say ,«„ .s negafve, let ,».—«.. ■^'"' «.(A-7) = <(A-/)' ^9) and ? is on the same side of each point, that is. its direction rom each point is the same. Also, since 'ts ^-stances rom t^ two points are inversely as their werghts, / must be nearest the point whose weight is greatest. 14 grassmann's space analysis. Case when w, -j~ '"^i = 0» or w, = — ;;/,.* — With this con- dition equations (7) and (8) become in,p, + inj^ = mXp, - A) = o .p, ( 10) and p —p^=p—j,^_ (u) Thus/ is in the same direction from each point, that is, not between them, and yet is equidistant from them. This re- quires either that the two points shall coincide, that is, /.^ =/i» which evidently satisfies (10) and (11); or else,/, and/^ being different points, that / shall be at an infinite distance. Thus the sum is in this case a point of zero w^eight at infinity.f Eq. (10) shows that a zero point at infinity is equivalent to a vector, or directed quantity, as stated in Art. i. It has been shown in Art. 2 that p^^p^ is the condition that /, and /, coincide; let us consider the equalit}' of weighted points in general, say m^p^ ^= vi„p^. Hence, by (7), there is found m^p^ — Wj/j = {in^ — m^p = o; hence, since /cannot be zero, w, — ni^ = o, or m, = in^ ; and therefore w,(/, —A) = O, or, since w,^o, /, — /„ =0, that is, /, =A- Therefore, if any two points are equal, their weights must be the same and their positions identical, that is, they are the same point. Exercise i. — To find the sum and difference of the two weighted points 3/, and/^' 3A+A = 4A 3A-A = 27, and the mean points are as shown in 2 _i_ _ 3 ^ f X "T ; t^i^ figure. The reciprocals of the -P' ^P' *^~' ^'distances of /, /„ and/' from /„, viz., i, ^, I", are in arithmetical progression, hence the points form a harmonic range. Exercise 2. — Given a circular disk with a circular disk of *Grassmann (1S62), Art. 222. f Compare the case of the resultant of unlike parallel forces of equal magnitude. SUM OF TWO WEIGHTED POINTS. 15 half its radius removed, as in the figure ; to find the centroid of the remaining portion. Take/, at center of large circle, /j at center of small circle, and p^ at the point of contact ; ( i\[ ^, ];, then /g = !(/>,-{- /'a)- The areas of the two cir- cles are as i 14; call them i and 4. Then it is as if there were a weight 4 at /,, and a weight — i at/3 ; hence J = [4/. - K/, + A)] - 3 = (7/. - A) -^ 6. Prob. I. Show that /,,/,, Jn^p^-\- m^p^, and m^p^ — 7n^p^ are four points forming a harmonic range. Prob. 2. An inscribed right-angled triangle is cut from a circular disk ; show that the centroid of the remainder of the disk is at the point (jTT — 2 sin 2a) /j — p^ sin 2« 3(7r — sin 20c) ' if /i is the center of the circle, p, the opposite vertex of the triangle, and a one of its angles. Art. 4. Sum of any Number of Points. As in the last article we assume the sum to be a point whose weight is equal to the sum of the weights of the given points ; thus, « « ^mp=p27u. (12) n Let e be some fixed point, and subtract e2m from both sides of (12) ; thus we have « « 2jn{p — e) ={p — e)2m, (13) 1 1 an equation which gives a simple construction (or p. n n If 2m = o, then 1n^ = — 2m, and « 2mp 2mp = m,p, + 2mp = m\ p, - -~ ], (14) 2m IG grassmann's space analysis. so that the sum becomes the difference of two unit points, or a vector whose direction is parallel to the line joining /, with the mean of all the other points of the system, and whose length is vi^ times the distance between these points. Since any point of the system may be designated as /,, it follows that the line joining any point of the system to the mean of all the others is parallel to any other such line. If '^mp = o, equation (14) shows that/, is the mean of all the other points of the system, and, since any one of the points may be taken as /,, any point of the system is the mean of all the others. Let ;/ = 3 in (12) and (13); then w,/, H- w,/, + m,p, = (;;/, + ;;/, + M,)p, (15) ;;/,(/. -e)-\- m^p-e) + mlp-e)={in,-^m^-^in^(p-e\ (16) and/ is on the line joining the point w^,/, + w^A ^^ith /j, and therefore inside the triangle p^p^p^ if the ihs are all positive. If W3 be negative and numerically less than m^-\-in^, then/ will have passed across the line /,/, to the outside of the tri- angle. If w, and m^ are negative and their sum numerically less than Wj, then / will have passed outside the triangle through /j, i.e., it will have crossed /^/j and /s/,. The point e must evidently always be in the plane /,/,/,. As a numerical example let in^ = 3, tn^ = 4, vi^=. — 5, so that (16) becomes p-e = |(/. -e)^ 2(A -e)- KA - e). Now, since e may be any point whatever, put e = p^; then 'p — p^ = -§.(/^ —A) + 2(A — A)' ^"^ ^^1^ construction is shown in the figure, p^ - p^ = f(/, -A), and 'p~p^ = 2(A - A)- As another example take/ = 4/, + 5A ~ ^A — ^A, or, by (13), making e =A, / -A = 4(A -A) + 5(A -A) - 2(A -/*) = A-A+A-A+7-A- SUM OF ANY NUMBER OF POINTS. 17 When any number of geometric quantities can be connected with each other by an equation of the form '2nip = o, in which the VIS are finite and different from zero, then they are said to be mutually dependent, that is, any one can be expressed in terms of the others. If no such relation can exist between the -5i' IV'- 'rip, ~Z1\ / quantities, they are independent. We obtain from what has preceded the following conditions; That two points shall concide, m,p,^in^p^ = 0. (17) That three points shall be collinear, mj, + ^n^P^ + ^>hP^ =0. (18) That four points shall be coplanar, ;//,/>. + ;;/,A + ^^hP. + ^^hP. = O. (19) It follows that three non-collinear points cannot be con- nected by an equation like (18) unless each coefificient is separately zero. Similarly four non-coplanar points cannot be connected by an equation like (19) unless each coefificient is separately zero. The significance of these statements will be presently illus- trated. The following are corresponding equations of condition for vectors : That two vectors shall be parallel, //,€, + ;/,e, :=0. (20) 18 GRASSMANX S SPACE ANALYSIS. That three vectors shall be parallel to one plane, ;/,e, + ;/,e, + ;^3e3 = O. (21) These conditions follow from the results of Art. 2, or from 'equations (17) and (18) by regarding the e's as points at infinity. If in addition to (21) we have '^ + ^h + ^h = o, (22) the extremities of the three vectors, if radiating from a point, -will be coUinear : for, let ^„ ... ^3 be four points so taken that .,?, — ^„ = 6, , ^2 ~ <^'o = ^2 ' ^3 ~ ^0 = ^3 ; then (21) becomes n,{e, — e^ + nle^ — e^ + ^3(^3 — e^ = O, ■or by (22) ;//, + /i,e^ + ;/3^3 = o, which by (18) requires e^, e„, c^ to be collinear. It may be shown similarly that ^ne =: ^n = o (23) are the conditions that four vectors radiating from a point shall have their extremities coplanar. Exercise 3, — Given a triangle e^e^e^ and a point / in its plane; pe^ cuts e^c„ in q^, pL\ cuts e„e^, in q^ , pe„ cuts e,e, in ^,, ^,^, cuts ^'/„ in /„, ,q^q, cuts e^e, in /, , and q,q^ cuts e^e^ in p„ : to show that Pa' Pi' 'I'lci p„ are collinear. Let/ = ;//„ + ;^,^,-f;/,^, ; jj\ then ^,, , ^, , q^ coincide re- spectively with ;//', -|- n„e^ , n^e^-\- n^e^, and n/a-\-n^€^ because p lies on the line joining e^ with q^ , etc. Hence, if -r„ , x^, y^ , j, are scalars, hence (;r„ -y,n,)e, -f (x, -j\n,)e, - n,(j^ -\-J\)e, = o. Now the ^'s are not collinear, and yet are connected by a SUM OF ANY NUMBER OF POINTS. 19 relation of the form of equation (i8); hence, as was there shown, each coefficient must be zero ; accordingly ^0 -}\'i, = '^r-Io'h =J'o -\-J\ = O, whence we find x^: x^ = «„ : — «,. hence (//„ — w,)/^ = '^o^o ~ ^'/i » ^"^i similarly {n—n^p, = n,€„ — n^t\, {j!, — n^)/>^ = n^e^ — n^e,. Adding, we have ('^> - ^i.)P. + kih - >io)P. + {n, - n)p„_ = o; therefore, by (iS), /„ '/i > ^2 ^^'^ collinear. Exercise 4. — Let / = ^Jie -=- 2fi be any point in the plane- 00 of the triangle e^e^e^ : show that lines through the middle points of the sides e^e^ , e^e^ , and e^e^ of the triangle parallel to e\p, c\p, and ^^p meet in a point P' = [(^^ + ''2)^0 + (^^2 + ^^o)^\ + («o + ^^,)^2] -^ 2:^«. By the conditions the vector from the middle point of e^e^ to p' is a multiple of the vector e\ — p ; hence / -K^^ + ^.) = A^o-p) or P' = K^. + ^2) + 4^0 -/) = i(^o + ^.) +J(^2 -/), or, substituting value of/, / = *(^, + ^',) + 4^„- -^//r-^2;/) = *(r„-f ^,)+j(^,-:^;^^H-:^«).- hence [(,r — i:)2n -f- 7/„(j — ;r)]r„ + w,(j/ — ,t')r, therefore, as in the previous exercise, each coefficient must be zero, whence ,i' = jf = -J, and substituting we find /' as above^ It follows also that the distances of p' from the middle points of the sides are the halves of the distances of/ from the oppo^ site vertices. 2 Prob. 3. Show that e = ^2e is collinear with p and />' of Exer- 20 GRASSMANN S SPACE ANALYSIS. cise 4. Also that, by properly choosing/, it follows that c is cul- linear with the common point of the perpendiculars from the vertices on the opposite sides, and the common point of the perpendiculars to the sides at their middle points. Prob. 4. Given two circles and an ellipse, as in the figure, with centers at '^ n. e: '>i. n. with similar values for e^ and e^. As a numerical example let the new reference triangle be formed by joining the middle points of the sides of the old one. Then e^ — \{c, -f rj, e^ = i(^„ + ^„), ^/ = ^{e, + e^) ; whence ^0 = - ^0' + e! + e^, e, = e^ - e/ + e,', e, = e,' + ^/ — ^./. Thus p = .r„r„ + ,r,r, + ,r/., = (- '^'o + '^'i + '^'.)^o' + U'o - 'I', + -l-',)^/ + (-S^o +^^ — ^.)^/- Exercise 6. — Three points being given in terms of the refer, ence points e^, i\, r„, find the condition that must hold between their weights when they are collinear. 2 2 2 Let />„ = 2/c, /i = ^;ne, p^ = '2ne\ then, k^, ^,, k^ being 000 scalars, we must have for collinearity. by (18), (3i> REFERENCE SYSTEMS. 33. that is, k^'2le -\- k'E^^ne -\- k^ne = o, whence {kj^ + k^7;i^ + /^2«o)^o + (^'oA + ^'i^«, + ^\«i)^i and, as e^, e^, l\ are not colhnear, the coefficients must be zero, by Art. 4 ; hence kj, + k^m^ + k^n^= kj^ -f /i-,///, + k^n^ = kj^ + /',;;/, + k.^n^ - o, and, by ehmination of the k's, /„ m, 11, which is the required condition of colhnearity. Prob. 7. If /> = 3^0 - ^, — ^2 . 4^«' = Zt\ + ^2 ' 4^:' = 3^2 + ^0 , 4''/ = y, + ^1 , show that 7/ = — 19^'/ — y\' + 29^/. 333 3 Prob. 8. Find the condition that four points ^ke, "Sle, 2i}ie, 2ne- 0000 shah be cophxnar. Ans. [/('„ , /j , ///, , n^] = o. Prob. 9. li p ^ 7ue^ -\- xe^ -{- ye„, and there exist between the scalars 70, x,y a hnear relation such as Atv + Bx -\- Cy = o, A, B, C being scalar constants, show that/ will always lie on a straight line which cuts the reference lines in Ae^ — Be^ , Ae^ — Ce^ , and Cfj — Be^. Consider the special cases when A =: B, B ^= C, C= A, A = B = C, A = o, B = o, and C = o. Prob. 10. If/ = 7i'e„ + Xi\ -\- ye„_ + se^, and there exist also an equation A7o -\- Bx -{- Cy -\- Dz = o, show that/ will lie on a plane c c which cuts the edges of the reference tetrahedron \\\ -^ ,", ° B A' e e —; -J-, etc. Also, if a second relation between the variables, such as A'tv + B' x + Cy -\- D'z = o, be given, then / lies on a line which pierces the faces of the reference tetrahedron in ^0 ^, ^. ABC A' B' a e c e DAB D' A' B' etc. '24 grassmann's space analysis. Art. 6. Nature of Geometric Multiplication* The fundamental idea of geometric multiplication is, that a product of two or more factors is that which is determined by those factors. Thus, two points determine a line passing through them, and also a length, viz., the shortest distance between them ; hence /j/', — L \s the sectf drawn from p^ to/^i or generated by a point moving rectilinearly from /^ to/^. The student should note carefully the difference between p^p^ and/^ — /"i ; they have the same length and direction, but the sect p^p^ is confined to the line through these two points, while the vector/, — />, is not. The sect has position in addi- tion to the direction and length possessed by the vector. Again, in plane space, two sects determine a point, the intersection of the lines in which they lie, and also an area, as will appear later, so that L^L^ — Pi i'l which p is not in general a unit point. In solid space, however, two lines do not, in general, meet, and hence cannot fix a point ; but two sects, in this case, determine a tetrahedron of which they are opposite edges. It appears, therefore, that a product may have different interpretations in spaces of different dimensions. Hence we will consider separately products in plane space, or planimetric products, and those in solid space, or stereometric products. Products of the kind here considered are termed " com- binatory," because two or more factors combine to form a new quantity different from any one of them. This is the fundamental difference between this algebra and the linear associative algebras of Peirce, of which quaternions are a special case. Before discussing in detail the various products that may arise, we will give a table which will serve as a sort of bird's-eye view of the subject. * Grassmann (1S44), Chap. 2 ; (1S62), Chap. 2. f See Art. i. NATURE OF GEOMETRIC MULTIPLICATION. 25 In this table and generally throughout the chapter we shall wsQ p, p^, p^, etc., for points; e, e^, e^, etc., for vectors ; L, L^, etc., for sects, or lines; ?/, z/^, etc,, for plane-vectors ; and P, P^, etc., for plane-sects, or planes. Also p, p^, etc., as used in this table will not generally be unit points. The products are arranged in two columns, so as to bring out the geometric principle of duality. Planimetric Products. /lAA = 'ire'^ (scalar). pL = area (scalar). /, • ^,4 = L. P.P.- P.P. ^P- P.P^ -P.P. •/'r,/'o=^ (area)Xscalar). L,L,=p. L^L^L^ = (area)''(scalar). Lp = area (scalar). A-/,A = A L^L^.L^L,.L,L,= (area)'(scalar) e^e^ = area (scalar). Stereometric Producis. P.P. = J- P,P, ^- L. P.P.P. = P- P.P.P.^P- PiP.P^P^ = volume (scalar). P,P„P,P, = (volume)' (scalar). pP = volume (scalar). Pp — volume (scalar). L^Li = volume (scalar). L^L^ = volume (scalar). pL = Lp = P. PL = LP = p. p.PA = P. P'P.P.=P- p.PAP,=L. P.p,p„j, = L. L-p.p,P,=p. L.P,P,P, = P. €.e, = ri. V.V, = e. €,€^63 = volume (scalar). 7j ^11^11^ = (volume)' (scalar). ^,e, . 636, = e. VJh-V,V. = V- 26 grassmann's space analysis. Laws of Combinatory Multiplication. — All combinatory products are assumed to be subject to the distributive law ex- pressed by the equation A{B -{- C) = AB ^ AC. The planimetric product of three pomts or of three lines, and the stereometric product of three points or planes, or of four points or planes, are subject to the associative law. That is, In Plane Space : AAA =AA-A=A-AA; AA4 = L^L^.L^ = L^ • L^L^. In Solid Space : P. P.P. =P.-P.P.= P.P. A ; P'.P.P. = P. ■ P.P. = P.P. • ^3. P.P.P.P^=P.-P.P.P^ —P.P.-PzP.'^ PPPP=P.PPP=PP PP, The commutative law of scalar algebra does not, in general, hold. Instead of this, in the products just given as being asso- ciative, a law prevails which may be expressed by the equation AB= -BA, from which it follows that the interchange of any two single factors of those products changes the sign of the product.* Since vectors are equivalent to points at co , the associative law holds for e^e^e^ and rjji^i]^- Art. 7. Planimetric Products. Product of Two Points.f— This has been fully defined in' Art. 6, and it is evident from its nature as there given that AA=-AA- (32) If p^ z=/>„ this becomes pj^ = o, which must evidently be true, since the sect is now of no length. Also, A(A -A) = AA - AA = AA- (33> * Grassmann (1862), Chap. 3. f Grassmann (1S62). Arts. 245, 246, 247. PLANIMETKIC PRODUCTS. 27 But/j — Px is a vector, say, e ; hence P,e=PxP-,'^ (34) ■lor the product of a point and a vector is a sect having the di- rection and magnitude of the vector ; or, again, multiplying a vector by a point fixes its position by making it pass through the point. To find under what conditions//'' will be equal to p^p^. Take any other point /j in the plane space under consideration, and write p = x,p, + x^^-^x,p,, p' =y,p, +y,p..+y,p,, with the conditions for unit points 2x = 2y = o. Then pp' = ,r, X, J, J'. AA + x^ x^ AA + x^ x^ J, J'3 Jz J> PzPv If this is to reduce to p^p„, we must have the third condition -^2^3 ~ '^'3^2 = '^'3^1 ~ '^'1 J3 = o> which requires that x\ = j^ = o, unless the coefificient of p^p^ is to vanish also. Thus pp' must be in the same straight line with/,/^- If, moreover, in addition -^J'2 — x^y^ = I' we shall have//' = p,p^. Hence //' is equal to/,/^ when, and only when, the four points are collinear, and /' is distant from/ by the same amount and in the same direc- tion that/2 is from/,. Product of Three Points. — By Art. 6 the product is what is determined by the three points. In solid space they would fix a plane, but, as we are now confined to plane space, this is not the case. The points evidently fix either a triangle or a parallelogram of twice its area, and the product p,p„p^ will be taken as the area of this, or an equivalent, parallelogram. This area is taken rather than that of the triangle, because it is what is generated by/,/, as it is moved parallel to its initial position till it passes through /j. We have p^P4>, =/, .p,p, = -/, .p,p, = - pj,p„ so that if we go around the triangle in the opposite sense the sign is changed. As this product possesses only the properties of mag- nitude and sisfii it is scalar. Write / = 2xp, /' = ^>/, /" =: :Ssp ; then 28 grassmanin's space analysis. ///' = 1 2 3 J, j» yz\p.P.P*'^ (35> that is, any triple point product in plane space differs from any other only by a scalar factor.* Finally, AAA = //A - A)(A - A) = A^e', (S^) if e = A — /. and e' ^^^ — /,. Product of Two Vectors. — Using the values of e and e' just given, we see that e and e' determine the same paral- lelogram that /,, A, and p^ do; hence the meaning of the product is the same in all respects in two-dimensional space. We shall have ee' = — e'e, for ee' = (A-A)(A-A)= -(A-A)(A-A)= -e'e; since we have shown that inverting the order changes the sign in a product of points. The result may be obtained also by regarding e and e' as points at infinity, or by consideration of a figure. As we have seen that ee' has, in plane space, precisely the same meaning 3.s p^p^p^ we may write P.P.P.^P.^^' ^ ee' = (A -/.)(A -A) =P.P.+P^P^+P.P.' {Z7) Thus the sum of three sects which form the sides of a triangle, all taken in the same sense as looked at from outside the triangle, is equal to the area of the triangle. Product of Two Sects. — Any two sects in plane space,. Z,, Z,, determine a point, the intersec- / tion of the lines in which they lie, and an area, that of a parallelogram as in T '"P\ ' the figure. Let /„ be the intersection, and take /, and ^ so that Z, =/„/■ ^""^ A = AA- The area * Grassmann (1S62), Art. 255. PLANIMETRIC PRODUCTS. 29- determined by L^ and L^ is then the same that we have given as the value oi p^p^p^. We write therefore ^:A = A/: 'P.P. = A/. A • A- (38) The third member of (38) is not to be regarded as derived from the second by ordinary transposition and reassociation of the points, for the associative law does not hold for the four points taken together, since A/i A -A = O- ^^^^ third member simply results from the definition of Z,Zj.* It may be taken as a model form which will be found to apply to several other cases, for instance to (38) when points and lines are inter- changed throughout. Thus, if/j = Z„/., and/, = L^L^we have /: A = L,L^ • L,L„_ = L^L^L^ . Z„. (39) For take// and //so that/,// = Z, and/^//= L,]p^p^ is evidently some multiple of L^ , say nL^ ; hence AA = «^o = ;^( AA • AA') • (AA • AAO = ^.(AA/,'-/,) • (/.AA' -A), by (38), = —2 • /.A/i'-AAA' • /.A. because /,/,// and PiPiPi ^^'^ scalar, = '-■ (P.P. • /■// • AA') • A- by (38), = L^L^L„ . Z„ , which was to be proved. Product of Three Sects. — The method has just been indi- cated, but we may also proceed thus: Let the lines be Z„, Z,, Zj, and let/^,/,,/„ be their common points. Take scalars «„, ;;, ?i^ so that Z„ = n^p^p^, etc., then L,L,L^ = nji^n, . p,p,- p.p^. p„p, = - n,n,n^ . p^p^ .p^p^ .p^p^ = - n,n,n^ • A/,A • A/o/> = '^o'^>«.(/',A,A)'- (40) * Grassmann applies the terms "eingewan it " and " regressiv " to a prod- uct of this kind, the first term being used in the Ausdehnungslehre of 1844, and the second in that of 1S62. See Chapter 3 of the first, and Chapter 3„ Art. 94, of the second. 30 grassmann's space analysis. Product of a Point and Two Sects, — Let/ be any point and let Z, and Z, be as in (38) ; then pL,L^ =p.p,p, .pj, =P -PoPJ^ -A =PoP,P^-PP,' (41) It has been here assumed that pL^L^ =/ . LJ^„. The prod- uct is not associative, for pL^ . L^ is the line Z„ times the scalar /Zj, a different meaning from that assigned in (41). As a rule, to avoid ambiguity, the grouping of such products will be indicated by dots. Product of Two Parallel Sects, — Let them be/je and np^e\ then, as in (38), /,e . np^e = n .p^e .p„e = n . ep, . e/, = n . ep,p^ . e, (42) that is, a scalar times the common point at 00 . Addition and Subtraction of Sects. — Let Z, and Zj be two sects, pg their common point, and p^ and p^ so taken that A =PoP^, A=AA; then Z, + Z, =/„/>, +AA = A(/: +A) = 2AA (43) / being the mean of/, and/„; hence the sum is that diagonal of the parallelogram which passes through /„. Also A- A=A(A-A)> (44) so that the difference of the two passes also through /„ and is parallel to the other diagonal of the parallelogram determined by Zj and Z„. If the two sects are parallel let them be w,/,e and n„p„e\ then «,/,€ + n^p^e = (;/,A + '^.A)e = {». + ».)pe,, (45) so that the sum is a sect parallel to each of them, having a length equal to the sum of their lengths, and at distances from them inversely proportional to their lengths. If «, = — «, the two sects are oppositely directed and of equal length, and the sum is nXp.e -p^e) = n^p, - P,)e, (46) which, being the product of two vectors, is a scalar area. PLANIMETRIC PRODUCTS. 31 Consider next /^ sects /,e, , /^e^ , . . . pne,,, and let c^ be some arbitrarily cliosen point; tlien n n 11 n n n ^p€ E e,^e - e^'Se + ^pe = e^:^^e -f- ^{p - ^Je. (47) Tlie second term of the tliird member of tliis equation, being a sum of double vector products, that is, a sum of areas, is itself an area, and is equal to the product of any two non-parallel vec- tors of suitable lengths. Therefore, a and (3 being such vec- tors, write ^'e = a and 2{p — i\)e = aft. Hence (47) become ^pe = cy, + a'/J = {e, - fS)a. (48) Let q be some point on the line ^pe\ then g^pe = O = qt\a + ga(3 ^ qt\a + Lxf3, by (37), hence qi\cx = — a(3 = f3a. The figure presents the geometrical mean- ing of the equation, and hence it appears that qa{=^ ^pe) is at a perpendicular distance from e, of cy£__^p- Oe Ta ~ T^e ' (49) It is easily seen that a sect possesses the exact geometrical properties of a force, namely, magnitude, direction, and position, and the discussion of the summation of sects which has just been given corresponds completely to the discussion of the re- sultant of a system of forces in a plane. In this algebra, then, the resultant of any system of forces is simply their sum, and this will be found hereafter to be equally true in three-dimen- sional space. The expression in (46) corresponds to a couple, as does also the ^X/ — ^'u)^ of (47); '^"^ t^'i's equation proves the proposition that any system of forces in a plane is equiva- lent to a single force acting at an arbitrary point, ^,,, and a couple. Equation (49) gives the distance of the resultant from this arbitrary point. Exercise 7. — To find x,y, .0 from the scalar equations 32 GRASSMANN S SPACE ANALYSIS. Multiply the equations by p^, p^, and p^ respectively, and add ; hence 3 3 3 3 x:^ap + }>:^hp -\- z:^cp — :^dp. 1 111 Now '^ap, ^bp, etc., are points : multiply the equation just written by '^ap.'^bp; thus z'^ap . ^bp:^cp = '2ap . :^bp . '2dp, because "^ap . 2ap = o, etc.; therefore z = ^ap . 2bp . :^dp -^ :^ap . ^bp^cp = [a, , b„^ , d,] H- [a, , b„_ , c,] , a very simple proof of the determinant solution. Of course X and }' will be found by multiplying by the other pairs of points. Exercise 8. — Forces are represented by given multiples of the sides of a par- allelogram ; determine their resultant. Let the parallelogram be double the triangle ^1 — -^3 — 2, /^2 = I ; 'incl wlien k^ = >^^ = i, ^'j, = /^^ = — 2. THE COMPLEMENT. 33 Prob. 12. There are given ;/ points /, . . ./„; to find a point e such that forces represented by the sects ^^ ^" Exercise 2 of Art. 3.) Prob. 14. Show that the relation of Prob. 13 holds for any four points whatever taken respectively on the four lines e/^, e^p, e/.^, e^p'. If the four points are all at the same distance from e^, show that the areas e^e^p, etc., become proportional to the sines of the angles between e/^ and i\p, etc. Art. 8. The Complement.* Taking point reference systems, or unit normal vector ref- erence systems, as in Art. 5, the product of the reference units taken in order being in any case unity, the complement of any reference unit is the product of all the others so taken that the unit times its complement is unity. To find the complements of quantities other than reference units the following properties are assumed : {a) The complement of a product is equal to tlie product of the complements of its factors. {b) The complement of a sum is equal to the sum of the complements of the terms added together. {c) The complement of a scalar quantity is the scalar itself. Considering now the point system in plane space f„, r,, e^ with the constant condition ^//^ = i, the sides of the refer- ence triangle taken in order are the complements of the oppo- site vertices, and vice versa. The complement of a quantity is indicated by a vertical line, as \p, read, complement of/. * See Ausdehnungslehre of 1862, Art. 8g. 34 grassmann's space analysis. Thus k„ = ^A, k/, = l(ko) = ^o» k2 = ^«^,, k„^. =l(i^O = ^,- For rj^o = ^//, =■ I, which agrees with the definition ; V,c^ = V,-V^ = i\i\ . e,t\ = — e^c^ . e,e^ = - ^//. .e^ = e„ by {a) and (38) ; ka^'/.= ko •\e,-\e^ = ^/.- ^.^0 • ^0^1 = (^o^/J' = I =^0^/., which agrees with (^) ; e^[c^ = ^//„ = o = rj^, = e^\e^. Next take any point /j = ^/r, and we have, by {b), lA=^v|.=/„./,+/..,r„+4.„.,=/y.4^^| - ^"jg - ^»j = L, (50) Thus the complement of a point is a hne,'^ which may be easily constructed by the fourth member of (50), winch ex- presses this line as the product of the points in which it cuts the sides e/^ and ^/, of the reference triangle. Comparing this equation with Ex. 3 in Art. 4, it appears that ]/, above is related to the point ^ % as the line/^/j of Ex. 3 is to the point '2)ic. Hence \p^ may be found by constructing this line cor- 1 g responding to ^ -, as shown in the figure of Ex. 3, Art. 4. / Again, the line |/, may be shown to be the anti-polar of p with respect to an ellipse of such dimensions, and so placed upon ^//„ that, with reference to it, each side of the reference triangle is the anti-polar of the opposite vertex.* F'rom this it appears that complementary relations are polar reciprocal relations. Take any point/"., = '^me, and we have "0 = .iVw = "^inc . ^'/\ £ = /., I/,, (51) *See Hyde's Directional Calculus, Arts 41-43 and 121-123. THE COMPLKMENT. 35 so tluit tliis product is commutative about the complement sign, and scalar. This is true of all such products when the quantities on each side of the complement sign are of the same order in the reference units. Take for instance the product /lAlAA- This is scalai because \p^p^ is a point, so that the whole quantity is equivalent to a triple-point product ; and we have/,/, I/3A = lAA -/.A = I KP^P^ l/iA) = A/4 i/iA. by {a) and (ct). If, however, such a quantity be taken 2iSp^p^ . \p^ it is neither scalar nor commutative about the sign | ; for, \p^ being a line, the product is that of two lines, that is, a point, and /.A • I A = - I A -/.A = - I (A • l/,A)- (52) Such products as we have just been considering are called by Grassmann "inner products,"* and he regards the sign | as a multiplication sign for this sort of product. Inasmuch, however, as these products do not differ in nature from those heretofore considered, it appears to the author to conduce to simplicity not to introduce a nomenclature which implies a new species of multiplication. For instance,/]^ will be treated as the combinatory product of p into the complement of g, and not as a different kind of product of/ into q. The term co-product may be applied to such expressions, regarded as an abbreviation merely, after the analogy of cosine for complement of the sine. Consider next a unit normal vector system. By the defini- tion we have l^= '2' \h= Khi) = — hy because zj ?, = i^i^ =■ i. Also, i Next let ;;/,/, -[- f'l^h and e„ = n^i^ -\- n^i^ * Gra'^smann (1S62), Chapter 4. 36 grassmann's space analysis. then, by {b) and {c), I e, = in^ 1 1, + ;//, 1 1^ = ;/z,z, — m,i^. (53) By the figure it is evident that | e, is a vector of the same length as e^ and perpendicular to it, or, in other words, taking the complement of a vector in plane space rotates it positively through 90°. The co-product e, j e^ is the area of the parallelogram, two of whose sides are e, and [e^ drawn outwards from a point; if e. is parallel to | e^ , this area vanishes, or eje^ = o; but, since I e., is perpendicular to e^ , e^ must in this case be perpendicular to e^ ; hence the equation ej 6, = o (54) is the condition that two vectors e, and e^ shall be perpendicu. lar to each other. The co-product e J e, , which will usually be written e,-, and called the co-square of e, , is the area of a square each of whose sides has the length Te^ ; hence ^e, =yij^=/i^ (55) Let a^ and a^ be the angles between i^ and e, and between z, and 62 respectively, as in the figure. Then e^e, = ;;/,«, — w,;^, = Te,Te„ sin {a.^ — a^, (56) the third member being the ordinary expression for the area of the parallelogram e,e^. Also = w,;/, + in^iK_ = Te, Te, cos {a, — «',), (57) the last member being found as before, remembering that sin (90° -\- a^— «j) = cos {a^ — a^). If in (57) we let e„ = e, , whence ;/, = ;;/, and n„ = vi, , we have 7e. = Ve^ = Vm: ^ m^. (58) \i Te, = Te., = i, then ;;/, — cos a^, w., = sin o',, «, = cos a„, n^ = sin a„ , and equations (56) and (57) give the ordinary trigo- nometrical formulas sin(«a'5 — a-,) = sin a^ cos a, — cos cr., sin a^, THE COMPLEMENT. 37 and cos {a^ — a-,) = cos o', cos 6^;,+ sin a^ sin a,. Squaring and adding (56) and (57), there results re, . T'e, = e.-e,? = (e,e,)^ + (e, | €,)\ (59) Attention is called to the fact, which the student may have already noticed, that such an equation as AB = AC, in which AB and AC a.re combinatory products, does not, in general, imply that B — C, for the reason that the equation A{B—C)=o can usually be satisfied without either factor being itself zero. Thus pL, — p^i means simply that the two quantities which are equated have the same magnitude and sign, which permits L.^ to have an infinity of lengths and positions, when p and L, are given. The equation />,/>, = A A » or/i(A — A) = O' A ^"d A being unit points, implies, however, that A =A' unless /^ is at 00 , that is, a vector. Exercise 9. — A triangle whose sides are of constant length moves so that two of its vertices remain on two fixed lines : find the locus of the other vertex. Let r„6j and r„e„ be the two fixed lines, and pp' p" the triangle. Let pe be per- pendicular to p' p" , /' — ^0 = -^'€1 and P" — ^0 = y^i ! then /" — p' = j'e„ — xe, , T{ye., — -re, ) = r = constant, by the con- ditions. Also, Tp'e = constant = 7/ic, say, and Tep = constant = 7ic, say. Hence e -p' = Tp'e. U{e -p') = ye, — xe^ = in{ye^ — xe,), T{ye, - xe,) and similarly/ — e = n\{ys^ — xe,). Therefore / — ^0 = p = ^e, 4- in{y€, — xe,) + ti\{ye, — xe,), an equation which, with the condition T{y€„ — xe,) = c, or y'e,^- - 2xye, \ e, + x'e,^ = c\ determines the locus to be a second-degree curve, which must in fact be an ellipse, since it can have no points at infinity. Let us rearrange the equation in p thus : p = x\{ I — vi)e, — 11 1 6,] -f j'[we, + 11 1 e,] — xe-\- ye, say, 38 GRASSMANN S SPACE ANALYSIS. SO that e = {i — in)e^ — n\e^ and e' = me^-\- n\e^\ then multi- ply successively into e and e'; therefore pe = ye'e and pe'^xee. Substituting these values of a- and j in the equa- tion of condition, we have el ■ {P^y + 26, 1 €,.pe . pe' + ejipej = r{ee'y, a scalar equation of the second degree in p. Exercise lO. — There is given an irregular polygon of ?i sides: show that if forces act at the middle points of these sides, proportional to them in magnitude, and directed all out- ward or else all inward, these forces will be in equilibrium. Let r„ be a vertex of the pol}-gon, and let 2e,, 2e„,. .. 26„ represent its sides in magnitude and direction. Then the mid- dle points will be ^„ -j- e, , f^-\- 2€, -}- 6„ , etc., and, using the complement in a vector system, we have 2/>e = (^+6,) 1 e,+(r„+2e,+e,) | e,+(^„+2e,+2e,+e3) \e,-\-.... + (^u + 26, + . . . + 26„_i + e„) I e„ Ve+>' 6^ + 26, ^e -f- 2 6„ ^^e-f-...4-2e„_i|6„ e -f- (-^ 6 = O, which was to be proved. Exercise ii. — A line passes through a fixed point and cuts two fixed lines; at the points of inter- section perpendiculars to the fixed lines are erected ; find the locus of the inter- section of these perpendiculars. Let the fixed lines be r„e, and c^e.,, and the fixed point (\ -f~ ^s ! the moving line cuts the fixed lines in /' and /". at which points perpendiculars are erected meeting in/. Let p — c^ = p, p' — e„ = xe, , p" — e, — je, , Te, = T^e, = I ; then p-= xe.^x' \6^=^ y6„-\-y' \€„ , whence pj e,= .r and p| e„= jj' EQUATIONS OF CONDITION, AND FORMULAS. 39 Also, since e^ -(- e^, p',j>" are collinear points, (-*'e, - e3)(je, — ej = o = xye^e^^ ye^e^ + X6^e^\ or, substituting values of x and y, P\e,. p\e^. e,e, 4-p|e, .e,e3+p|c .€36, =0, an equation of the second degree in p, and hence representing a conic. Prob. 15. If a, b, c are the lengths of the sides of a triangle, prove the formula a^ =l b" -\- r — 2bc cos A, by taking vectors e,, e^, and 65 — ei equal to the respective sides. Prob. 16. If ^„6j and e^e-i. are two unit lines, show that the vec- tor perpendicular from e^ on the line {e^ + ^^i)(^o + ^^2) is ^^^1^2 , ,, ■ X . , . , , , , . abe^e, . [bei — ae), of which the length is -— *-? — -. From this derive the Cartesian expression for the perpendicular from the origin upon a straight line in oblique coordinates, ab ?,\r\ CfO -4- {a'' + b' — 2ab cos go)^, go being angle between the axes. Prob. 1 7.- If three points, we^ + ne^, me^ -{- ne,, me. + ne^, be taken on the sides of the reference triangle, then the sides of the complementary triangle, | {me^ + ne^), etc., will be respectively paral- lel to the corresponding sides of the triangle formed by the assumed, points {me^ + Jie^), {me^ + "f^^), etc. Art. 9. Equations of Condition, and Formulas. Several equations of condition are placed here too-ether for convenient reference : some have been already given ; others follow from the results of Arts. 7 and 8. When we have /■A = o, the two points coincide ; or 2np = O, I the three points are collinear ; e,e, = o, or L,L, = o, or «,Z, -|- n^L^ = o, the two lines coincide ; L,L,L, = o, or 'SnL = o, 1 the three lines are confluent. w,e, -f ;/,e, = o, (62) (60). (61) J the two vectors are parallel (points at infinity coincide); e, I e. = o, (63) 40 grassmann's space analysis. the two vectors are perpendicular ; either point lies on the com- plementary line of the other. L,\L, = o, (64) either line passes through the complementary point of the other. If we write the equation P = -^1^1 + -^'.e,, x^e^ is the projection of p on ei parallel to e,, and x^e^ is the projection of p on e^ parallel to e^. Multiply both sides of the equation into e^; therefore pe^ = x^e^e^, or x^ = pe^ -i- e^e^. Similarly, multiplying into e^, we have pe, = x^e^e^, or x^ = pe, -7- e„ei, whence p = + . (65) The two terms of the second member of (65) are therefore the projections of p on e, parallel to e^, and on e^ parallel to e,, respectively.* Let e, and e, be unit normal vectors, say, t and |z; then (65) becomes p = t. p\t — \i . pi = I . p\i -\- ip . |z; (66) or, if Zj and i^ be used instead of i and 1 1, P= ^^'P\h + h-P\h' (67) Again, in (65) let p = e^, clear of fractions, and transpose ; therefore e,e, . €3+ e.e,. e, + €36, .e, = O, (68) a symmetrical relation between any three directions in plane space. Let T^e, = Te^ = Te^ — • i, and multiply {6'S) into | €3, thus €,6, + e^ej.eje, + 636, .eje, = O, (69) which is equivalent to sin («- ± /3) = sin a cos yS ± cos a sin ft, the upper or lower sign corresponding to the case when €-3 is * Grassmann (1844), Chapter 5 (1862), Art. 129. Hyde's Directional Calcu- lus, Arts. 46 and 47. EQUATIONS OF CONDITION, AND FORMULAS. 41 L±„ between e^ and e,, or outside, respectively. Writing in (69) ] e^ instead of e,, we have e, I e^ — ej 63 . 63! e, + €36, . e^e^ = O, (70) which gives the cos {a ± /3). These formulas being for any three directions in plane space, are independent of the magni- tude of the angles involved. There is given below a set of formulas for points and lines, arranged in complementary pairs, and all placed together for convenient reference, the derivation of them following after. /=(AAA)"'[A -/AA + A -/AA + A -/AoAl- ) L={L,L,Ly[L, . LL,L, + Z. . LL^L, + 4 . ZZ„ZJi ' p={pj.p.y\\p.p.-p\p. +\p.p.-p\p.^ i pj.-p\p^, P.P.-PJ\ = - A -AAA + P.'P.P.P, = P^'P.P.P,— P.'PlP2p.^ 31, L,\M, M, L,\M„_ LAM, L,\M, A 1^0 Ak> a!^» /. \g. PA ^1 /. I Qi A 1^0 Aki PMi The complementary formula to {jj^ is not given, but may l>e obtained by putting Z's and M's for /'s and ^'s. Derivation of Equations (7i)-(77). — Equation (71). Write p = x^p^ -\- x,p, -\- x^p^, and multiply this equation by p,p^ ; then p.p^p = x,p,p,p„ or x, = pp,p, -^ p,p,p,. Multiplying similarly by p^p, and by p^p,, we find ^1 = PP^Po -^ P.P.P. and x^= pp,p,-^p,p.p,. The substitu- AA- ki = PMx9^ = A A I ^i^s — — A AU A Ak k7 1 Al^i 2 PM-i > ^2 kl ^2 I ^2 L,LAM,= L.AM,M,= L,LAM,M,= P.PlP-X ■ ^0^.^2 = (71) (72) (73) (74) (75) (76) (77) 43 GRASSMANN S SPACE ANALYSIS, tion of these values gives the first of (71), and the second is similarly obtained or may be found by simply putting Z's for /'s in the first. Equation (72). Write/ = x^ \p,p, + x\ \ p,p, + x, \ p,p^, and multiply into \p^ ; thus/|/„ = ^\P^Pxpy Find in the same way values of x^ and x„, and substitute. Equation (73). Write />,/„ .p^p^ ~ xp^ -\- ypi-, and multiply by//, ; therefore//, .p,p,.p,p, = xpp.,p,, or, by Eq. (38), P.PP.-P.P^P. = -'-PP.Px = - ^P-.PP:^ ox,x=- p,p,p,. Multiply- ing by//, we find y — p^p.p,, and on substituting obtain the first of {JT)). For the second put /,/, .p^p^ = xp^ + j/4, and. proceed in a similar way. Equation (74). In the first of (73) put /^^p^ = |^,. Equation (75). In the fourth of (73) put -^1-^2 ^^^ Pit ^%^^^ l^l» ^4 ^ 1^2- Equation {^6). Multiply (75) by/,. Equation ij"]). In the first of (72) put q., for/, and multiply by A/. ^.-^0^1 ; then /"o/.A • QS.q-^. = ^.?, \P^p2 • ^. \Po+ 9.9 ^ \P.P. • Qi 1/.+ ^0^7, 1 A/*. • q^ I A Ak. p\q. Al^. Ai^o A!^i +Ak. + Ak. Ai^o Al^i A 1^0 A,'^> Al^n A'^, by (76), which is equivalent to the third order determinant of equation {77)-^ Exercise 12. — To show the product of two determinants as a determinant of the same order. Let /„ = ^le, /, = 2vtc, /, = 'S7ie, q^ = 2Xe, q^ = 2 pie, q., = 2ve; then A/, A = [^0- '^^' '^]' q,q,q. = [^c /^,» n] ; also A ko = ^0^0 + A^, + ^.^.' /, ko = ^'^o'^o + '«.^. + ^^^X., etc. Sub- stituting these values in (77), we have the required result. A solution may also be obtained directly without the use of {77). 2 Let the ^'s be as above, but write A = ^^'-/'i = 2wq,p.^ = 2nq. Then p^p^p^ = 2/q.2P!q.2uq=[/„, m^, n,']q„q,q, = [/„ ;«,, «,][A„, ;y„ rj. * Grassmann (1S62), Art. 173. EQUATIONS OF CONDITION, AND FORMULAS. 43 Also /„ = /„:SA^ + i;2}xe + l^'2ve with similar values for/, and/^j which on being substituted in P«P\P% g^^^ '^'^^ result. Equation {jj), however, exhibits the product in a very compact, symmetrical, and easily remembered form.* Exercise 13. — Show that the sides/,/^? AA' A/^i of the tri- angle /j/^A cut the corresponding sides 1/3, |/j, j/^ of the com- plementary triangle in three collinear points. The three points of intersection are, using (74), /A-lA==-A-AlA+A-AlA.AA-lA = -A-AlA+A-AiA. A/i • !A = ~ A -/i IA +/! • AIA ' of which the sum is zero, showing that the points are collinear. It may be shown in the same way that the lines joining corresponding vertices are confluent. Exercise 14. — If the sides of a triangle pass through three fixed points, and two of the vertices slide on fixed lines, find the locus of /\ the other vertex. / \ _^ p y \ „ Let the fixed points and lines be ^^3 7><^\' /,, /„ ^3, Z,, U, and /, p', p" the ^,-'"'"/ \V vertices of the triangle, as in the ^' \ figure. Then p'p^p" ^o\ p' coin- ^ cides with pp^.L^ and/'' with pp„. L„\ hence substituting (//, . I^^PX^i- Pip) — O, the equation of the locus, which, being of the second degree in p, is that of a conic. Prob. 18. Show that if the three fixed points of the last exercise are collinear, then the locus of / breaks up into two straight lines. Use equation (73). Prob. 19. If the vertices of a triangle slide on three fixed lines, and two of the sides pass through fixed points, find the envelope of the other side. (This statement is reciprocally related to that of Exercise 14, that is, lines and points are replaced by points and * These methods may be applied to determinants of any order by using a space of corresponding order. 44 grasSiMANn's space analysis. lines respectively, and the resulting equation will be an equation of the second order in Z, a variable line.) Prob. 20. Show that if the three fixed lines of Problem 19 are confluent, then the envelope of L reduces to two points and the line joining them. Art. 10. Stereometric Products. The product of two points in solid space is the same as in plane space. See Art. 7. Product of Three Points. — Any three points determine a plane, and also, as in Art. 7, an area ; hence /^/../j is a plane-sect or a portion of the plane fixed by the three points whose area is double that of the triangle p^p^py It may be shown, in the manner used in Art. 7 for the sect, that no plane-sect, not in this plane, can be equal to/,/„/'3, and that any plane-sect in this plane having the same area and sign will be equal to/j/^/'j.* Of course /',/'„/'3 is not now scalar. Product of Four Points. — Anj' four non-coplanar points determine a tetrahedron, sav ii^ ^ip^ . ^ y ""--.^ p^p.p^p^, and six times the vol- "\^ ume of this tetrahedron is £ \ / ?^^,' ^^ taken for the value of the Jy "v, y' product, because this is the ^^* volume of the parallelepiped generated by the product /,/,/3, — i.e. the parallelogram/^,/,, — when it moves parallel to its initial position from/, to /^. Let P.-P. = e, P,-P. = e, p,-p, = e", then /.AAA = P.P.P.^" = P^P.^'^" = A^e'e". (78) 3 .! 3 3 If /, =:^ke, p„ = '^le, p^ = '^me, /, = :^ne, then " p,p.p,p, - '^ke^le'2inc'^ne = [/-„, /,, w,,, n^ . i\e^c^e^ ; (79) from which it appears that any two quadruple products of points differ from each other only by a scalar factor, that is, they differ only in magnitude, or sign, or both ; hence such products arc themselves scalar.f If /.AAA = O' ^^^ volume of the tetrahedron vanishes, so that the four points are coplanar. * Grassmann (1862), Art. 255. \ Grassmann (1S62), Art. 263. STEREOMETRIC PRODUCTS. 45 Product of Two Vectors. — The two vectors determine an area as in Art. 7, but they also determine now a plane direc- tion, so that the product e^e^ is a plane-vector, and is not scalar as in plane space. Also, e^e^ differs from p^e.e^ now just as e differs from pe; namel}', e,e^ has a definite area and plane direction, that is, toward a certain line at infinity, whWo. p^e^e^ is fixed in position by passing through/,. Equation (ly) there- fore does not hold in solid space. Product of Three Vectors. — Three vectors determine a parallelepiped as in the figure above, and ee'e" is therefore the volume of this parallelepiped. Any other triple vector product can differ from this only in magnitude and sign. For let 616,63 be such a product, and write 333 6 = ,r,e, -f- x^e^-\- x^e^ — ^xe, e' = ^ ye, e" = '^ze ; then 1 1 1 ee'e" = '2xe'2ye'2,ze = x^ x^ x^ fi y. y. 6. 6^63, (80) so that the two products only differ by the scalar determinant factor. Hence the product of three vectors must be itself a scalar, by Art. i. Since, then, the product of four points has precisely the same signification as that of three vectors, we may v/rite P.PJJ. =p,ee'e" = ee'e" = (A - AXA - /.)(/4 " A) = P.P.A - /s/^/i + /./,/. - AAP,- (81) Thus the sum of the plane-sects forming the doubles of the faces of a tetrahedron, all taken positively in the same sense AS looked at from outside the tetrahedron, is equal to the volume of the tetrahedron. Compare equation (37). If ee'e" = O, the volume of the parallelepiped vanishes, and the three vectors must be parallel to one plane. Product of Two Sects. — In solid space two sects determine a tetrahedron of which they are opposite edges. Thus AAAA =/'A. • AA = ^.A = AA -AA = AZ„ (82) so that the stereometric product of two sects is commutative, and has the same meaning as that of four points. 46 grassmann's space analysis. Product of a Sect and a Plane-Sect. — Let them be L and P, and let /„ be their common point; take p^, p^, p^ so that L—p^p^ and P = p„p.2p^- L and P evidently determine the point /„, and also the parallelepiped of which one edge is L and one face is P, so that the product should be made up of these two factors. Hence we write LP = pj, . /'„AA = Pop J J, . Po'y PL = pjj, . A/, = AAA/, -/o = LP. ) ^^^^ If L is parallel to P, p^ is at infinity, and, replacing it by e, (83) becomes PL = LP= ep, . ep,p^ = epj,p, . e. (84) Product of Two Plane-Sects. — Let them be P^ and P,^, and let L be their intersection, while/j and A are such points that P, = Lp^ and P^ = Lp^\ then P^ and P^ determine the line L and also a parallelepiped of which they are two adjacent faces, and P.P., = Lp, . Lp, = Lpj^ .L=- P,P,. (85) If Pj and P., are parallel, L is at infinity, and is equivalent to a plane-vector, say to ?/ ; hence, substituting in (84), PA = vp, . vp. = np.P. • '/ = - P.P.- (86) Product of Three Plane-Sects. — By (85) and (83) this must be the square of a volume times the common point of the three planes; or, if /„' /i' A:' A t>e taken in such manner that P. =PJ.P.^ P. =AAA' P. ^P.P.P.^ then P,P,P,^ 021. oil .012 =023.0123.01 ={p,p,pJ,Y .p,\ (87) the suffixes being used instead of the corresponding points. If />„ be at infinity, the three planes are parallel to a single line, and may be written P, = n^ep^p^, etc., and then treated as above. Product of Four Pla-ne-Sects.* — Let the planes be /*„ . . . P^, and let/„ . . . />, be the four common points of the planes taken three by three. w„ . . . tt^ may be so taken that P^ = '^o/,AA' .£tc. ; then P.P^A, = n,n.n.^ii, . 123 . 230 . 301 . 012 = n^i'.'iMPoPJJ^.y- (88) * Grassmann (1S62), Art. 300. STEREOMETRIC PRODUCTS. 47 Product of Two Plane- Vectors. — Let //^ and -q^ be two plane- vectors or lines at infinity ; let e be parallel to each of them, and €1 and e^ so taken that ?/, = ee^, i]^ = ee^, then ViV, = e^i • ee, = ee^e, . e = — 7/^7/,, (89) because 7/^ and 7/^ determine a common direction e, and a paral. lelepiped of which three conterminous edges are equal to €, ej, €2, respectively. Product of Three Plane-Vectors. — Take €^, e^, €3 so that '/.V.V3 = » ' e,e, . €36. .€,€, = ii{e,e^ej. (90) The directions e, . . . €3 are common to the plane-vectors ?/j . . . //s taken two by two. Several conditions are given here together which follow from the results of this article. Two points coincide. Three points coUinear. AAAA =/.A-AA = Z^Z, = o, Four points co planar ; two lines intersect. e^e„ = o, Vectors parallel. 616263 = O, Three vectors parallel to one plane. PA = o, (91) Two planes coincide. P.P.P. = o, (92) Three planes collinear. P P P P = P P .P P = L,L, = o, (93) Four planes confluent ; two lines intersect. V.V, = O, (94) Plane-vectors parallel. 'AV^^s = 0, (95) Three plane-vectors parallel to one line. Sum of Two Planes. — Let them be P^ and P^, let Z be a sect in their common line, and take />, and/, so that /*, =: Z,/,, P^ = Lp^ ; then /^. + P, = Z(/. + A) = 2Lp, (96) p being the mean of /^ and p^. Also z'. -Z'^^AA-A); (97) whence the sum and difference are the diagonal plane through Z, and a plane through Z parallel to the diagonal plane which is itself parallel to Z, of the parallelepiped determined by P^ 48 grassmann's space analysis. and P^. If TP,= TP^, P, ± P^ will evidently be the two bisecting planes of the angle between them. The bisecting planes may also be written Yp^Yp °' P^TP^^P-^TP,, (98) If the two planes are parallel, let /; be a plane-vector parallel to each of them, that is, their common line at infinit)-, and let/, and p^ be points in the respective planes; then we may write P^ = n^pj}, P^ = '^sA^' whence ^, + A = k^hp. + ^I'.P^v = ('^ + ''J/v- (99) If «, -|- 11^ = o, this becomes ^: + A = «.(A-A)'A (100) the product of a vector into a plane-vector and therefore a scalar, by (80). Two plane-vectors may be added similarly, since they will have a common direction, namely, that of the vector parallel to both of them. Exercise 15. — If two tetrahedra e^e^e^e^ and e^e^e^e^ are so situated that the right lines through the pairs of corresponding vertices all meet in one point, then will the corresponding faces cut each other in four coplanar lines. The given conditions are equivalent to ^// . e^e^ = o ■^ e e ' e c ' ^^ e e ' . e c ' ^^ e c ' . c e ' ^ e e ' .€ c ' '=■ e e ' € e ' . Two of the intersecting lines of faces are ^//^ . ^„V,V/ and £'/„c^ . ^/r„V/, and, if these intersect, we must accordingly have, by (93), 012 . o'i'2' . 123 . i'2'3' = o = 012 . 123 . o'i'2' . i'2'3' = 0123 . o'i'2'3' . I2i'2', the last factor of which is equivalent to the fourth condition above, since quadruple-point products in solid space are associative. Similarly all the other pairs of intersections may be treated. Exercise 16. — The twelve bisecting planes of the diedral angles of a tetrahedron fix eight points, the centers of the inscribed and escribed spheres, through which they pass six by six. The sum and difference of two unit planes are their two STEREOMETRIC PRODUCTS. 49 bisecting planes, by (97). Let the tetrahedron be e^e^e^e^, and let the double areas of its faces be A^ = Ti\e^e^, etc. ; then a pair of bisecting planes will be " ,' ' ±_ —~ or e^clA^c\ ± ^363)- The pair through the opposite edge will be e^cj^A^e^ ± ^,^,)' If there be a point through which the six internal bisecting planes pass, it must be on the intersection of these two planes taken with the upper signs, and we infer by symmetry that it 3 must be the point '^Ae. Another internal bisecting plane is e^e^jyA^e^ -\- A^c^, which gives zero when multiplied into '^Ae^ as do also the other three. To obtain all the points we have only to use the double signs, so that they are ± A^e^ ± A^c^ ±_ A^l\ ±_ A^e^. This gives eight cases, namely, + + + + -+ + + + + + - + + + +-+ + + + -++ +- + - The eight apparent cases that would arise by changing all the signs are included in these because the points must be essen- tially positive. Moreover, no positive point could have three negative signs, because the sum of any three faces of the tetra- hedron must be greater than the fourth face. It will be found on trial that six of the bisecting planes will pass through 2( ± Ae) with any one of the above arrangements of sign. Prob. 21. The twelve points in which the edges of a tetrahedron are cut by the bisecting planes of the opposite diedral angles fix eight planes, each of which passes through six of them. Prob. 22. The centroid of the faces of a tetrahedron coincides with the center of the sphere inscribed within the tetrahedron whose vertices are the centroids of the respective faces of the first tetrahedron. Prob. 23. If any plane be passed through the middle points of two opposite edges of a tetrahedron, it will divide the volume of the tetrahedron into two equal parts. 50 grassmann's space analysis. Art. 11. The Complement in Solid Space. According to the definitions of Art. 8 the complementary relations in a unit normal vector system are as follows : hh= 1(10 = z, I ^3^ = 1(10 = h r- (loi) I, = Kl. '•s'-i ' = hi,, ^-^^(^^. Let e = ^li ; then I e = /,hh + ^.'s^, + ^.hh = j{Ki, - /.z.)(/,^ - /,0. (I02) so that |e is a plane-vector. The figure, which is drawn in isometric projection, shows that the two vectors /./., — /„/j i,i;. Vs* f^ ^"<^ O3 ~ ^s'l' whose prod- uct is /, • I e, are both perpen- dicular to e; for the first is perpendicular to /,;, -^ /„;.,, which is the orthogonal pro- jection of eupon ZjZ^,and to Zj, and therefore is also per- pendicular to e, while the second is perpendicular to /^z, -|- l^i^ and to z.^, and therefore to e. Hence | e is a plane-vector perpendicular to e ; and, since |(|e) = e, the converse is also true, i.e. the complement of a plane-vector is a line-vector normal to it. The figure shows that e is equal to the vector diagonal of the rectangular parallelepiped whose edges have the lengths /,, /„ /,, hence Te = 4//7 + 1: + 1;. (103) Multiply equation (102) by e; therefore el 6 = (/,z, + Kj„_ + /^'^V/,^/, + Kr^i, + /3^0 = /;+C + /3^=7'V = ei, (104) so that the co-square of a vector is equal to the square of ita tensor. The product e|e is that of a vector e into a plane- vector perpendicular to it, as has just been shown ; it is there- THE COMPLEMENT IN SOLID SPACE. 51 fore a volume which is equivalent to Te . T\e\ hence, by (104), eje= Te . T\e ■= T^e, or 7^6= T\e. Hence, the complement of a vector in solid space is a plane-vector perpendicular to it and having the same tensor, or numerical measure of magni- tude.* 3 Let a second vector be e' ^^ini ; then 1 6 1 e' = /,///j -\- /„7/V„ -\- I^m^ == e' I e. (105) Now e\e' being the product of e into the plane-vector ] e', is the volume of the parallelepiped in the fig- ure, that is, TeTe' sin (angle between e and | e') = ZeT^e'cosf. Hence e\e' — e' \ e=l^in^~\-l^vi„-\-l^i!i^= TeTe cos f. (106) If Te— Te' — \, l^ . . . l^, w, , . . 111^ are di- rection cosines, and (105) gives a proof of the formula for the cosine of the angle between two lines in terms of the direction cosines of the lines. We have also in this case ee = (/,;//, - /„;//,) 1 1^ + {J^m^ — l^m^ \ i^ -f {l^m^ — l,in^) \ /„, and, taking the co-square, {eej= (sin :')= = (/,;//,- l,my+{hn-l,vi:)' + {l^m-l^my. (107) If e\e' =0, (108) 6 is parallel to the plane-vector perpendicular to e', that is, e is perpendicular to e' , as is also shown by (106). Let ?/ = I e, if = I e' ; then r]\jf =■ 1 e . e' = e' I e = e| e' = TeTe' cos f = TfjTif cos ^'. (109) and 7;|r/' = O (l lO) is the condition of perpendicularity of two plane-vectors. Also either e|7/ = o, or //'|e = o, (nO is the condition that a vector shall be perpendicular to a plane- vector, for the first means that e is parallel to a vector which is * Grassmann (1862), Art. 335. GRASSMANN S SPACE ANALYSIS. perpendicular to ?/, and the second that ?/ is parallel to a plane- vector which is perpendicular to e. Equations (7i)-(77) of Art. 9 become stereometric vector formulae if e,, e^, etc., be substituted for />,,/,, etc., and ?;, , y^. etc., for Zj, Z,, etc. For instance, {jG) gives the vector formulas -12 I 1 i > ViV, I 7i V, (112) ^/J^/ '/J '7, For lack of space no treatment of the complement in a point system in solid space is given. Exercise 17. — To prove the formulas of spherical trigo- nometry cos a = cos d cos <: -|- sin d sin c cos A, and sin a sin d _ sin c sin A sin i? sin 6" Take three unit vectors e, , e^, e^ parallel to the radii to the vertices of the spherical triangle, then «=:(angle bet. e„ and e^), A—{a.ugle bet. e^e^ and e^e^), etc. In eq. (112) put e^e^ for e/e„'; hence e^e, | e^e^ = sin d sin c cos A = e-. e^\e^ — e^ | e^ . ej €3 = cos a — cos d cos c. Again, ^(e^e, . 6,63) = 7\e,e.^e3 . e,) = Te^e^e, = T(e^6^ . e,e-,)= T{€,e.^ . e.e,)', or sin i? sin c sin ^i = sin a sin whence den 2 r(f/e,e,+ f/e,e3) the ordinary value sin s sin (s — a) sin I? sin c Expanding, the numerator becomes i 4- Ue^eJ Ue^e^, and the denominator |'^2(i -\- 6"e,eJ Ue^e^). Also there is obtained Ue e\Ue 6 = ' -'^J^- dent. The remainder is left to the stu- i^u-' ^1^3 Prob. 24. If e^ , €„, €3, drawn outward from a point, are taken as three edges of a tetrahedron, show that the six planes perpen- ADDITION OF SECTS IN SOLID SPACE. 53 dicular to the edges at their middle points all pass through the end of the vector p = -Vr( I ^.^3 • ^r+ I ^36, . e/ + | e^e, . e,'). (Sug- gestion. We must have (p — ^ej | e, = o, with two other similar expressions.) Prob. 25. Show that e, | ee' and ee' . \ e are three mutually per- pendicular vectors, no matter what the directions of e and e' may be. Prob. 26. Let e,, e^, e^ be taken as in Prob. 24 ; let A^ be the area of the face of the tetrahedron formed by joining the ends of these vectors, and 2A^ = Te^e^, etc.; also 6*, = Angle between e,e, and 6,^3, etc.: then show that we have the relation, analogous to that of Prob. 15, Art. 8, ^;^= A^"--^AJ-^A^''— 2A^A^ cos 6,— 2A^A, cos 6^— 2A^A^ cos 0^. If ^'j ... 6*3 are right angles, this becomes the space-analog of the proposition regarding the hypotenuse and sides of a right-angled triangle. (Suggestion. 2A, = r(e., - ej{e^ - e,).) Prob. 27. There are given three non-coplanar lines t'_,e, , t^,,e^, ^^63; planes cut these lines at right angles, the sum of the squares of their distances from c„ being constant. Show that the locus of the common point of these three planes is {p\e^y-\-{p\e^)'-\-{p\e^y — c'', if Te, = Te., = re, = i. Art. 12. Addition of Sects in Solid Space. Two lines in solid space will not in general intersect, so that their sum will not be, as in eq. (43), a definite line. For let /}^e^ and/^e^ be any two sects: then A^i + Ae, = />,e^ +Ae. + ^'„(e, + ej — ^„(e, + e,) = ^o(e, + e.J + (A - Oe, + (/, - ^„)e,; that is, the sum is a sect passing through an arbitrary point e^, and a plane-vector, the sum of the two in the equation. The sum cannot be a single sect unless the two are coplanar ; for let />, =/, + A'^i + je^ -j- .ce^, 63 being a vector not parallel to eje^; hence /.e, + p,e^ = />,€, + (/, + xe^ + j/e, + ^63)6, = P. (e, + e,) + xe^ (e, + eJ + 5-636, = (/, + ^ej (e, + eJ + ^£36, ; and this cannot reduce to a single sect unless 2 = 0, that is, un- less /^e, and/^e^ are coplanar. Since a plane-vector is a line at 54 grassmann's space analysis. CO , the sum of two lines may always be presented as the sum of a finite line and a line at co . If the sum of any two sects is equal to the sum of any other two, their products will also be equal, that is, the two pairs will determine tetrahedra of equal volumes. For let L^-\- L^ — L^-\- L^\ then squaring we have L^L^ = L^L^ , since L^L^ = o, etc. An infinite number of pairs of sects can be found such that the sum of each pair is equal to the sum of any given pair; for let a given pair be p^e^ -\- p.2^„, and take a new pair {xj^ + ^'.A)(".e, + ?'.ej -t- {y,p, + y,p,){v,e, + v^e^) {x\n^ '^}\t\)p,e„_ + {xji, +J',e\)/,6,. This will be equal to the given pair if we have '*'i^^i+ jr^. = ^',^^2 4- A"^. =1. and x,u^ +y,v, = x^n^ ^-y,v, = o). Since there are eight arbitrary quantities with onl}' four equations of condition, the desired result can evidently be ac- complished in an infinite number of ways. Let /,6, ./^e^ . . . . pn€„ be n sects, and let 5" be their sum, and ^„ any point, then S=2pe = e^2e - e„2e + ^pe = e^:Ee + ^{p - e,)e, .... (i 13) 1 the sum of a sect and a plane-vector as before. If 2{p — e\)e is parallel to ^e it may be written as the prod- uct of some vector e' into 2e, that is, e'2e, when the sum be- comes 5 = t\^6 4" e'^e = (.?, -{- e')^e, a sect, because ^„ -|- e' is a point. In no other case does 5 reduce to a single sect. If 26 = O. 5 becomes a plane-vector. Of the two parts compos- ing S, the sect will be unchanged in magnitude and direction if e^ be moved to a new position, while the plane-vector will in general be altered. It is proposed to show that a point (/ may be substituted for e„ such that the plane-vector will be perpen- dicular to ^'e. Writing 5 = ^^6 -iq - e;)2e -f 2{p - e,)e, and, for brevity, putting q — c^ = p, ^e = a, 2{p — e^e = \/3, so that S = qa — pa-\-\^, (114) ADDITION OF SECTS IN SOLID SPACE. 55 we must have for perpendicularity, by (in), {\/3 — pa) j ar = o = I /Sa — pa . \ a, or pa .\a^a. p\a — p . a- ^\/!ia. i^^S) The second member is obtained from the first by substitut- ing in eq. (74) p ior p^ and a for /'^ and q^, in accordance with the statement at the end of Art. ii. If in (115) we make p|« = o,. p will be the vector from t\ to q taken perpendicularly to a, say p, = \afi-^ a'-=q^ — e,. (116) Since a and (3 are known, the required point has been found. Multiply (115) by a; then, using (75), — ap . a- = pa , ai = „e„ = 5 to its normal form. -S" = ^0(^1 + ej + (/, — r„)e, + (A - ^o)^2- For convenience suppose/, and /„ to be taken at the ends of the common per- *Grassmann (1862), Art. 346. 56 grassmann's space analysis. pendicular on /,e, and/^e^, and moreover let ^o = il/i H-Zs). /, — ^j = i = — (/j — e^; then z|e, = i|e, = o. Accordingly >^^ ^ ^Xe. + e^) +z(6. - ej = ^(e.+0+ ^''+'-_|;^y "'^ . |(6. + 6,) Hence the normal form of 5 is Exercise 20. — Forces are represented by the six edges of a tetrahedron c^e^, e^c.^, c^e^, e^e^, e^e^, e^e^\ find the 5, reduce to normal form, and consider the special case when three diedral angles are right angles. 5 = ej^e^ + ^, + ^3) + ^2^3 + ^3^1 + ^^e„ = ^o(e,+ea+e3)+(^2-^.)(^3-^>)= ^o(e,+e,+e3)+'(e,-e,Xe3-e,) = ^o(e, + e, + 63) + 6,63 + €36, + e,e, , in which e, = c\ - e, , etc. Hence o- (. _!_ {e,e, + 636. 4- e,e,) | (e. + e^ + e,)^ , ^ , ^ n ^ - ^^0 + (iH^-^:T^37 ^^ ' + ^ + ^^ For the rectangular tetrahedron let e^ =: ai^ , e„ = ^f, , 63 = CZ3 , /, , z^, Zg being unit normal vectors. Then we find + ^+7^17-! (^^■ + '^'= + ^'3). Exercise 21. — A pole 50 feet high stands on the ground and is held erect by three guy-ropes symmetrically arranged about it, attached to its top and to pegs in the ground 50 feet from the pole. The wind blows against the pole with a pressure of 50 pounds in the direction e^ — p, when e^ is at the bottom of ADDITION OF SECTS IN SOLID SPACE.' 5!? the pole, and / divides the distance between two of the pegs in the ratio — : find the tension on the guys and the pressure on the ground. Evidently only two of the guys will be in tension ; let their pegs be at '_^__ 25 in -\- n in V2 n V2 Vi/i^ -\- ;/' — inn y and z being the tensions, and x -\- w the upward pressure. Prob. 28. Three equal poles are set up so as to form a tripod, and are mutually perpendicular; a weight 7C' hangs upon a rope which passes over a pulley at the top of the tripod, and thence 3 down under a pulley at the ground at a point / = ^le, in which 1 ^, . . . ^3 are at the feet of the poles, and -^7 = i ; if the rope is pulled 58 grassmann's space analysis. so as to raise w, show that the pressures on the poles, supposing the pulleys frictionless, are Prob. 29. Six equal forces act along six successive edges of a cube which do not meet a given diagonal; show that if the edges of the cube be parallel to i,, i„, i^, and F be the magnitude of each force, then .S == — 2F\ (i, + ^^ + ^s)* i^ the diagonal taken be parallel to i, + h-\- h- Prob. 30. Three forces whose magnitudes are i, 2, and 3 act along three successive non-coplanar edges of a cube; show that the normal form of ^ is -^ = (^0 + H^ + i^. - tVO(^ + 2l._+ 3h)+ AK^, + 2l, + 31,). Prob. 31. Forces act at the centroids of the faces of a tetrahedron, perpendicular and proportional to the faces on which they act, and all directed inwards, or else all outwards; show that they are in equilibrium. INDEX. Addition of points, page 9. of weighted points, 12-16. of vectors, 12. of sects (planimetric), 30. of sects (stereometric), 53. of plane-sects, 47. Coincidence of two points, 17, 39, 47. Collinearity of three points, 17, 39, 47. Coplanarity of four points, 17, 39, 47. CoUinearity of ends of three vectors, 18. Coplanarity of ends of four vectors, 18. Combinatory products, 24. multiplication, Laws of, 26. Complement, ^7,, 50. Condition, equations of, 17, 39, 47. Co-product, 35. Co-square, 36. Difference of points, 10. Difference between p^p^ and p2~P\, 24. Determinants, product, 42. Equations of condition, 17, 39, 47. Formulae for points and lines in plain space, 41. for vectors in solid space, 52. Geometric multiplication, 24. Inner product, 35. Inscribed and escribed spheres, centers. Laws of combinatory multiplicdtion, 26. Mean point, 13. Multiplication, Geometric, 24. Combinatory, 24. Parallelism of vectors, 47. of plane-vectors, 47. Perpendicularity, Condition of, 36, 51. Plane sects. Product, 46. vectors. Product, 47. parallel, 47. three parallel to one line, 47- Planimetric products, 25, 26. Point at infinity= vector, 9. Product, Combinatory, 24. Products of two points, 26. of three points, 27. of two vectors, 28. of two sects, 28. of three sects, 29. of two determinants, 42. Reference systems, 20. Scalar, definition, 9. Sect, definition, 8. Sects, products, 28, 29, 45. products of parallel, 30. planimetric sum, 31. stereometric sum, 53. Spheres, inscribed and escribed, centers of, 48. Spherical trigonometry, formulae, 52. Tcnsor= T, 9. Unit= U, 9. Vector, Definition of, 10. plane, 45. Vector products and conditions, 17, 18, 47- Wrench, 55. SHORT-TITLE CATALOGUE OP THE PUBLICATIONS OP JOHN WILEY & SONS, New York. London: CHAPMAN & HALL, Limited. ARRANGED UNDER SUBJECTS. Descriptive circulars sent on application. Books marked with an asterisk (*) are sold at net prices only, a double asterisk (.**) books sold under the rules of the American Publishers' Association at net prices subject to an extra charge for postage. All books are bound in cloth unless otherwise stated. AGRICULTURE. 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Svo, 3 00 Prescott and Winslow's Elements of Water Bacteriology, with Special Refer- ence to Sanitary Water Analysis i2mo, i 25 * Reisig's Guide to Piece-dyeing Svo, 25 00 Richards and Woodman's Air, Water, and Food from a Sanitary Stand- point Svo, 2 00 Richards's Cost of Living as Modified by Sanitary Science i2mo, i 00 Cost of Food, a Study in Dietaries i2mo, i 00 * Richards and Williams's The Dietary Computer Svo, i 50 Ricketts and Russell's Skeleton Notes upon Inorganic Chemistry. (Part I. Non-metallic Elements.) Svo, raorocco, 75 Ricketts and Miller's Notes on Assaying Svo, 3 00 Rideal's Sewage and the Bacterial Purification of Sewage Svo, 3 50 Disinfection and the Preservation of Food Svo, 4 00 Rigg's Elementary Manual for the Chemical Laboratory Svo, i 25 Robine and Lenglen's Cyanide Industry. (Le Clerc.) Svo, Rostoski's Serum Diagnosis. (Bolduan.) lamo, i 00 Ruddiman's Incompatibilities in Prescriptions Svo, 2 00 * Whys in Pharmacy i2mo, i 00 Sabin's Industrial and Artistic Technology of Paints and Varnish Svo, 3 00 Salkowski's Physiological and Pathological Chemistry. (Orndorff.) Svo, 2 50 Schimpf 's Text-book of Volumetric Analysis l2mo, 2 50 Essentials of Volumetric Analysis i2mo, i 25 * Qualitative Chemical Analysis Svo, i 25 Spencer's Handbook for Chemists of Beet-sugar Houses i6mo, morocco, 3 00 Handbook for Cane Sugar Manufacturers i6mo, morocco, 3 00 Stockbridge's Rocks and Soils Svo, 2 50 * TiUman's Elementary Lessons in Heat Svo, i 50 * Descriptive General Chemistry Svo, 3 00 Treadwell's Qualitative Analysis. (Hall.) Svo, 3 00 Quantitative Analysis. (Hall.) Svo, 4 00 Turneaure and Russell's Public Water-supplies Svo, 5 00 Van Deventer's Physical Chemistry for Beginners. (Boltwood.) i2mo, 1 50 * Walke's Lectures on Explosives Svo, 4 00 Ware's Beet-sugar Manufacture and Refining Small Svo, cloth, 4 00 Washington's Manual of the Chemical Analysis of Rocks Svo, 2 00 Wassermann's Immune Sera: Haemolysins, Cytotoxins, and Precipitins. (Bol- duan.) i2mo, I 00 Well's Laboratory Guide in Qualitative Chemical Analysis Svo, i 50 Short Course in Inorganic Qualitative Chemical Analysis for Engineering Students i2mo, i 50 Text-book of Chemical Arithmetic i2mo, i 25 Whipple's Microscopy of Drinking-water Svo, 3 50 Wilson's Cyanide Processes i2mo, 1 50 Chlorination Process i2mo, i 50 Winton's Microscopy of Vegetable Foods Svo, 7 50 Wulling's Elementary Course in Inorganic, Pharmaceutical, and Medical Chemistry i2mo, 2 00 5 CIVIL ENGINEERING. BRIDGES AND ROOFS. HYDRAULICS. MATERIALS OF ENGINEERING. RAILWAY ENGINEERING. Baker's Engineers' Surveying Instruments lamo, 3 00 Bixby's Graphical Computing Table Paper igi X 24i inches. 25 ** Burr's Ancient and Modern Engineering and the Isthmian Canal. (Postage, 27 cents additional.) 8vo, 3 50 Comstock's Field Astronomy for Engineers 8vo, 2 50 Davis's Elevation and Stadia Tables 8vo, i 00 Elliott's Engineering for Land Drainage i2mo, i 50 Practical Farm Drainage i2mo, i 00 ♦Fiebeger's Treatise on Civil Engineering 8vo, 5 00 Folwell's Sewerage. (Designing and Maintenance.) 8vo, 3 00 Freitag's Architectural Engineering. 2d Edition, Rewritten 8vo, 3 50 French and I /es's Stereotomy 8vo, 2 50 Goodhue's Municipal Improvements i2mo, i 75 Goodrich's Economic Disposal of Towns' Refuse 8vo, 3 50 Gore's Elements of Geodesy. 8vo, 2 50 Hayford's Text-book of Geodetic Astronomy 8vo, 3 00 Bering's Ready Reference Tables (Conversion Factors) i6mo, morocco, 2 50 Howe's Retaining Walls for Earth i2mo, i 25 Johnson's (J. B.) Theory and Practice of Surveying Small 8vo, 4 00 Johnson's (L. J.) Statics by Algebraic and Graphic Methods 8vo, 2 00 Laplace's Philosophical Essay on Probabilities. (Truscott and Emory.). i2mo, 2 00 Mahan's Treatise on Civil Engineering. (1873.) (V/ood.). 8vo, 5 00 * Descriptive Geometry Svo, i 50 Merriman's Elements of Precise Surveying and Geodesy '^vo, 2 50 Merriman and Brooks's Handbook for Surveyors i6mo, moro^- ~ 00 Nugent's Plane Surveying Svo, 3 ^.^ Ogden's Sewer Design i2mo, 2 00 Patton's Treatise on Civil Engineering Svo half leather, 7 50 Reed's Topographical Drawing and Sketching 4to, 5 00 Rideal's Sewage and the Bacterial Purification of Sewage Svo, 3 50 Siebert and Biggin's Modern Stone-cutting and Masonry. Svo, i 50 Smith's Manual of Topographical Drawing. (McMillan.) Svo, 2 50 Sondericker's Graphic Statics, with Applications to Trusses, Beams, and Arches. Svo, 2 00 Taylor and Thompson's Treatise on Concrete, Plain and Reinforced Svo, 5 00 * Trautwine's Civil Engineer's Pocket-book i6mo, morocco, 5 00 Wait's Engineering and Architectural Jurisprudence Svo, 6 00 Sheep, 6 50 Law of Operations Preliminary to Construction in Engineering and Archi- tecture 8vo, 5 00 Sheep, 5 50 Law of Contracts 8vo, 3 00 Warren's Stereotomy — Problems in Stone-cutting Svo, 2 50 Webb's Problems in the Use and Adjustment of Engineering Instruments. i6mo, morocco, i 25 Wilson's Topographic Surveying Svo, 3 50 BRIDGES AND ROOFS. Boiler's Practical Treatise on the Construction of Iron Highway Bridges. .8vo, * Thames River Bridge 4to, paper, Burr's Course on the Stresses in Bridges and Roof Trusses, Arched Ribs, and Suspension Bridges 8vo, 6 Burr and Falk's Influence Lines for Bridge and Roof Computations. . . .8vo, 3 00 Design and Construction of Metallic Bridges 8vo, 5 00 Du Bois's Mechanics of Engineering. Vol. II Small 4to, 10 00 Foster's Treatise on Wooden Trestle Bridges 4to, 5 00 Fowler's Ordinary Foundations 8vo, 3 50 Greene's Roof Trusses Svo, I 25 Bridge Trusses Svo', 2 50 Arches in Wood, Iron, and Stone Svo, 2 50 Howe's Treatise on Arches Svo, 4 00 Design of Simple Roof-trusses in Wood and Steel Svo, 2 00 Johnson, Bryan, and Turneaure's Theory and Practice in the Designing of Modern Framed Structures Small 4to, 10 00 Merriman and Jacoby's Text-book on Roofs and Bridges: Part I. Stresses in Simple Trusses Svo, 2 50 Part II. Graphic Statics Svo, 2 50 Part III. Bridge Design Svo, 2 50 Part IV. Higher Structures Svo, 2 50 Morison's Memphis Bridge 4to, 10 00 Waddell's De Pontibus, a Pocket-book for Bridge Engineers. . i6ino, morocco, 2 00 Specifications for Steel Bridges i2mo, i 25 Wright's Designing of Draw-spans. Two parts in one volume Svo, 3 50 HYDRAULICS. Bazin's Experiments upon the Contraction of the Liquid Vein Issuing from an Orifice. (Trautwine.) Svo, 2 00 Bovey's Treatise on Hydraulics Svo, s 00 Church's Mechanics of Engineering Svo, 6 00 Diagrams of Mean Velocity of Water in Open Channels paper, i 50 Hydraulic Motors Svo, 2 00 Coffin's Graphical Solution of Hydraulic Problems i6mo, morocco, 2 50 Flather's Dynamometers, and the Measurement of Power i2mo, 3 00 Folwell's Water-supply Engineering Svo, 4 00 Frizell's Water-power Svo, 5 00 Fuertes's Water and Public Health i2mo, i 50 Water-filtration Works i2mo, 2 50 Ganguillet and Kutter's General Formula for the Uniform Flow of Water in Rivers and Other Channels. (Hering and Trautwine.) Svo, 4 00 Hazen's Filtration of Public Water-supply Svo, 3 00 Hazlehurst's Towers and Tanks for Water-works Svo, 2 50 Herschel's 115 Experiments on the Carrying Capacity of Large, Riveted, Metal Conduits Svo, 2 00 Mason's Water-supply. (Considered Principally from a Sanitary Standpoint.) Svo, 4 00 Merriman's Treatise on Hydraulics Svo, 5 00 * Michie's Elements of Analytical Mechanics Svo, 4 00 Schuyler's Reservoirs for Irrigation, Water-power, and Domestic Water- supply Large Svo, 5 00 ** Thomas and Watt's Improvement of Rivers. (Post, 44c. additional.). 4to, 6 00 Turneaure and Russell's Public Water-supplies Svo, 5 00 Wegmann's Design and Construction of Dams 4to, 5 00 Water-supply of the City of New York from 1658 to 1895 4to, 10 00 Williams and Hazen's Hydraulic Tables Svo, i 50 Wilson's Irrigation Engineering Small Svo, 4 00 Wolff's Windmill as a Prime Mover Svo, 3 00 Wood's Turbines Svo, 2 50 Elements of Analytical Mechanics Svo, 3 00 7 MATERIALS OF ENGINEERING. Baker's Treatise on Masonry Construction 8vo, Roads and Pavements 8vo, Black's United States Public Works Oblong 4to, * Bovey's Strength of Materials and Theory of Structures 8vo, Burr's Elasticity and Resistance of the Materials of Engineering 8vo, Byrne's Highway Construction 8vo, Inspection of the Materials and Workmanship Employed in Construction. i6mo, Church's Mechanics of Engineering 8vo, Du Bois's Mechanics of Engineering. Vol. I Small 4to, ♦Eckel's Cements, Limes, and Plasters Svo, Johnson's Materials of Construction Large Svo, Fowler's Ordinary Foundations Svo, * Greene's Structural Mechanics Svo, Keep's Cast Iron Svo, Lanza's Applied Mechanics . .Svo, Marten's Handbook on Testing Materials. (Henning.) 2 vols Svo, Maurer's Technical Mechanics Svo, Merrill's Stones for Building and Decoration Svo, Merriman's Mechanics of Materials Svo, Strength of Materials i2mo, Metcalf's Steel. A Manual for Steel-users lamo, Patton's Practical Treatise on Foundations Svo, Richardson's Modern Asphalt Pavements Svo, Richey's Handbook for Superintendents of Construction i6mo, mor., Rockwell's Roads and Pavements in France i2mo, Sabin's Industrial and Artistic Technology of Paints and Varnish Svo, Smith's Materials of Machines i2mo. Snow's Principal Species of Wood Svo, Spalding's Hydraulic Cement i2mo, 2 00 Text-book on Roads and Pavements i2mo, 2 00 Taylor and Thompson's Treatise on Concrete, Plain and Reinforced Svo, s 00 Thurston's Materials of Engineering. 3 Parts Svo, 8 00 Part I. Non-metallic Materials of Engineering and Metallurgy Svo, 2 00 Part II. Iron and Steel Svo, 3 50 Part III. A Treatise on Brasses, Bronzes, and Other Alloys and their Constituents Svo, 2 50 Thurston's Text-book of the Materials of Construction Svo, 5 00 Tillson's Street Pavements and Paving Materials Svo, 4 00 Waddell's De Pontibus. (A Pocket-book for Bridge Engineers.). .i6mo, mor., 2 00 Specifications for Steel Bridges i2mo, 1 25 Wood's (De V.) Treatise on the Resistance of Materials, and an Appendix on the Preservation ef Timber Svo, 2 00 Wood's (De V.) Elements of Analytical Mechanics Svo, 3 oo Wood's (M. P.) Rustless Coatings: Corrosion and Electrolysis of Iron and Steel Svo, 4 00 . RAILWAY ENGINEERING. Andrew's Handbook for Street Railway Engineers 3x5 inches, morocco, I 25 Berg's Buildings and Structures of American Railroads 4to, 5 00 Brook's Handbook of Street Railroad Location i6mo, morocco, I 50 Butt's Civil Engineer's Field-book i6mo, morocco, 2 50 Crandall's Transition Curve i6mo, morocco, 1 50 Railway and Other Earthwork Tables Svo, i 50 Dawson's "Engineering" and Electric Traction Pocket-book. . i6mo, morocco, 5 00 8 5 00 5 00 5 00 7 50 7 50 5 00 3 00 6 00 7 50 6 00 6 00 3 50 2 50 2 50 7 50 7 50 4 00 5 00 5 00 I 00 2 00 5 00 3 00 4 00 I 25 3 00 I 00 3 50 Dredge's History of the Pennsylvania Railroad: (1879) Paper, 5 00 * Drinker's Tunnelling, Explosive Compounds, and Rock Drills. 4to, half mor., 25 00 Fisher's Table of Cubic Yards Cardboard, 25 Godwin's Railroad Engineers' Field-book and Explores' Guide. . . i6mo, mor., 2 50 Howard's Transition Curve Field-book i6mo, morocco, i 50 Hudson's Tables for Calculating the Cubic Contents of Excavations and Em- bankments 8vo, I 00 Molitor and Beard's Manual for Resident Engineers i6mo, i 00 Nagle's Field Manual for Railroad Engineers i6mo, morocco, 3 00 Philbrick's Field Manual for Engineers l6mo, morocco, 3 00 Searles's Field Engineering i6mo, morocco, 3 00 Railroad Spiral i6mo, morocco, 1 50 Taylor's Prismoidal Formulae and Earthwork 8vo, 1 50 * Trautwine's Method of Calculating the Cube Contents of Excavations and Embankments by the Aid of Diagrams 8vo, 2 00 The Field Practice of Laying Out Circular Curves for Railroads. izmo, morocco, 2 50 Cross-section Sheet Paper, 25 Webb's Railroad Construction i6mo, morocco, 5 00 Wellington's Economic Theory of the Location of Railways Small 8vo, s 00 DRAWING. Barr's Kinematics of Machinery. . . .• 8vo, * Bartlett's Mechanical Draviring , 8vo, * " " " Abridged Ed 8vo, Coolidge's Manual of Drawing 8vo, paper Coolidge and Freeman's Elements of General Drafting for Mechanical Engi- neers Oblong 4to, Durley's Kinematics of Machines 8vo, Emch's Introduction to Projective Geometry and its Applications 8vo, Hill's Text-book on Shades and Shadows, and Perspective 8vo, Jamison's Elements of Mechanical Drawing 8vo, Advanced Mechanical Drawing 8vo, Jones's Machine Design: Part I. Kinematics of Machinery 8vo, Part II. Form, Strength, and Proportions of Parts 8vo, MacCord's Elements of Descriptive Geometry 8vo, Kinematics; or. Practical Mechanism 8vo, Mechanical Drawing 4to, Velocity Diagrams ". 8vo, MacLeod's Descriptive Geometry Small 8vo, * Mahan's Descriptive Geometry and Stone-cutting 8vo, Industrial Drawing. (Thompson.) 8vo, Moyer's Descriptive Geometry 8vo, 2 00 Reed's Topographical Drawing and Sketching 4to, 5 00 Reid's Course in Mechanical Drawing 8vo, 2 00 Text-book of Mechanical Drawing and Elementary Machine Design. Svo, 3 00 Robinson's Principles of Mechanism 8vo, 3 00 Schwamb and Merrill's Elements of Mechanism 8vo, 3 co Smith's (R. S.) Manual of Topographical Drawing. (McMillan.) Svo, 2 50 Smith (A. W.) and Marx's Machine Design 8vo, 3 00 Warren's Elements of Plane and Solid Free-hand Geometrical Drawing. i2mo, i 00 Drafting Instruments and Operations i2mo, 1 25 Manual of Elementary Projection Drawing i2mo, i 50 Manual of Elementary Problems in the Linear Perspective of Form and Shadow i2mo, i 00 Plane Problems in Elementary Geometry i2mo, i 25 9 3 00 3 00 5 00 4 00 I 50 I 50 I 50 3 50 Warren's Primary Geometry i2mo, 75 Elements of Descriptive Geometry, Shadows, and Perspective 8vo, 3 5» General Problems of Shades and Shadows 8vo, 3 00 Elements of Machine Construction and Drawing 8vo, 7 50- Problems, Theorems, and Examples in Descriptive Geometry Svo, 2 50 Weisbach's Kinematics and Power of Transmission. (Hermann and Klein.) Svo, 5 Oq, Whelpley's Practical Instruction in the Art of Letter Engraving lamo, 2 oo- Wilson's (H. M.) Topographic Surveying Svo, 3 50- Wilson's (V. T. ) Free-hand Perspective Svo, 2 50 Wilson's (V. T.) Free-hand Lettering Svo, I oo- Woolf' s Elementary Course in Descriptive Geometry Large Svo, 3 00 ELECTRICITY AND PHYSICS. Anthony and Brackett's Text-book of Physics. (Magie.) Small Svo, 3 00 Anthony's Lecture-notes on the Theory of Electrical Measurements. . . . i2mo, i 00 Benjamin's History of Electricity Svo, 3 00 Voltaic Cell Svo, 3 00 Classen's Quantitative Chemical Analysis by Electrolysis. (Boltwood.).8vo, 3 00 Crehore and Squier's Polarizing Photo-chronograph Svo, 3 00 Dawson's "Engineering" and Electric Traction Pocket-book. i6mo, morocco, 5 00 Dolezalek's Theory of the Lead Accumulator (Storage Battery). (Von Ende.) i2mo, 2 50 Duhem's Thermodynamics and Chemistry. (Bi*gess.) Svo, 4 00 Flather's Dynamometers, and the Measurement of Power i2mo, 3 00 Gilbert's De Magnete. (Mottelay.) .Svo, 2 50 Hanchett's Alternating Currents Explained i2mo, i 00 Bering's Ready Reference Tables (Conversion Factors) i6mo, morocco, 2 5a Holman's Precision of Measurements Svo, 2 00 Telescopic Mirror-scale Method, Adjustments, and Tests. .. .Large Svo, 75 Xinzbrunner's Testing of Continuous-current Machines Svo, 2 00 Landauer's Spectrum Analysis. (Tingle.). . ' Svo, 3 00 Le Chateliers High-temperature Measurements. (Boudouard — Burgess.) i2mo, 3 00 Lr.b's Electrochemistry of Organic Compounds. (Lorenz.) Svo, 3 00 * Lyons'j Treatise on Electromagnetic Phenomena. Vols. I. and II. Svo, each, 6 00. * Michie's Elements of Wave Motion Relating to Sound and Light Svo, 4 00 Niaudet's Elementary Treatise on Electric Batteries. (Fishback.) i2mo, 2 50 * Rosenberg's Electrical Engineering. (Haldane Gee — Kinzbrunner.). . .Svo, i 50. Ryan, Norris, and Hoxie's Electrical Machinery. Vol. I Svo, 2 50 Thurston's Stationary Steam-engines Svo, 2 50 * Tillman's Elementary Lessons in Heat Svo, i 50 Tory and Pitcher's Manual of Laboratory Physics Small Svo, 2 00 Ulke's Modern Electrolytic Copper Refining Svo, 3 00 LAW. * Davis's Elements of Law Svo, 2 50 * Treatise on the Military Law of United States Svo, 7 00 * Sheep, 7 50 Manual for Courts-martial i6mo, morocco, i 50 Wait's Engineering and Architectural Jurisprudence Svo, 6 00 Sheep, 6 so Law of Operations Preliminary to Construction in Engineering and Archi- tecture 8vo 5 00 Sheep, 5 50 Law of Contrasts Svo, 3 00 Winthrop's Abridgment of Military Law I2m0i 2 50 10 MANUFACTURES. Bernadou's Smokeless Powder — Nitro-cellulose and Theory of the Cellulose Molecule i2mo, 2 50 Holland's Iron Founder i2mo, 2 50 " The Iron Founder," Supplement i2mo, 2 50 Encyclopedia of Founding and Dictionary of Foundry Terms Used in .the Practice of Moulding lamo, 3 00 Eissler's Modern High Explosives 8vo, 4 00 Effront's Enzymes and their Applications. (Prescott.) 8vo, 3 00 Fitzgerald's Boston Machinist i2mo, i 00 Ford's Boiler Making for Boiler Makers i8mo, i 00 Hopkin's Oil-chemists' Handbook 8vo, 3 00 Keep's Cast Iron 8vo, 2 50 Leach's The Inspection and Analysis of Food with Special Reference to State Control Large 8vo, 7 50 Matthews's The Textile Fibres 8vo, 3 50 Metcalf's SteeL A Manual for Steel-users i2mo, 2 00 Metcalfe's Cost of Manufactures — And the Administration of Workshops. 8vo, 5 00 Meyer's Modern Locomotive Construction 4to, 10 00 Morse's Calculations used in Cane-sugar Factories i6mo, morocco, i 50 * Reisig's Guide to Piece-dyeing 8vo, 25 00 Sabin's Industrial and Artistic Technology of Paints and Varnish 8vo, 3 00 Smith's Press-working of Metals 8vo, 3 00 Spalding's Hydraulic Cement i2mo, 2 00 Spencer's Handbook for Chemists of Beet-sugar Houses i6mo, morocco, 3 00 Handbook for Cane Sugar Manufacturers i6mo, morocco, 3 00 Taylor and Thompson's Treatise on Concrete, Plain and Reinforced 8vo, 5 00 Thurston's Manual of Steam-boilers, their Designs, Construction and Opera- tion 8vo, 5 00 * Walke's Lectures on Explosives 8vo, 4 00 Ware's Beet-sugar Manufacture and Refining Small 8vo, 4 00 West's American Foundry Practice i2mo, 2 50 Moulder's Text-book i2mo, 2 50 Wolff's Windmill as a Prime Mover 8vo, 3 00 Wood's Rustless Coatings : Corrosion and Electrolysis of Iron and Steel. .8vo, 4 00 MATHEMATICS. Baker's Elliptic Functions 8vo, i 50 * Bass's Elements of Differential Calculus i2mo, 4 00 Briggs's Elements of Plane Analytic Geometry i2mo, i 00 Compton's Manual of Logarithmic Computations i2mo, i 50 Davis's Introduction to the Logic of Algebra 8vo, i 50 * Dickson's College Algebra Large i2mo, I 50 * Introduction to the Theory of Algebraic Equations Large i2mo, I 25 Emch's Introduction to Projective Geometry and its Applications 8vo, 2 5a Halsted's Elements of Geometry 8vo, i 75 Elementary Synthetic Geometry 8vo, i 50 Rational Geometry i2mo, i 75 * Johnson's (J. B.) Three-place Logarithmic Tables: Vest-pocket size. paper, 15 100 copies for 5 00 * Mounted on heavy cardboard, 8X10 inches, 25 10 copies for 2 00 Johnson's (W. W.) Elementary Treatise on Differential Calculus . . Small 8vo, 3 00 Johnson's (W. W.) Elementary Treatise on the Integral Calculus. Small 8vo, i 50 11 Johnson's (W. W.) Curve Tracing in Cartesian Co-ordinates i2mo, i oo Johnson's (W. W.) 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Vector Analysis and Quaternions, by Alexander Macfarlane. No. 9. Differential Equations, by William Woolsey Johnson. No. 10. The Solution of Equations, byl Mansfield Merriman. No. 11. Functioas of a Complex Variable, by Thomas S. Fiske. Maurer's Technical Mechanics 8vo, 4 00 Merriman and Woodward's Higher Mathematics 8vo, 3 00 Merriman's Method of Least Squares Svo, 2 00 Rice and Johnson's Elementary Treatise on the Difierential Calculus. . Sm. Svo, 3 00 Differential and Integral Calculus. 2 vols, in one Small Svo, 2 50 Wood's Elements of Co-ordinate Geometry Svo, 2 00 Trigonometry: Analytical, Plane, and Spherical i2mo, i 00 MECHANICAL ENGINEERING. MATERIALS OF ENGINEERING, STEAM-ENGINES AND BOILERS. 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Iron and Steel 8vo, 3 50 Part III. A Treatise on Brasses, Bronzes, and Other Alloys and their Constituents 8vo, 2 50 Text-book of the Materials of Construction 8vo, 5 00 Wood's (De V.) Treatise on the Resistance of Materials and an Appendix on the Preservation of Timber 8vo, 2 00 13 Wood's (De V.) Elements of Analytical Mechanics 8vo, 3 00 Wood's (M. P.) Rustless Coatings: Corrosion and Electrolysis of Iron and Steel 8vo, 4 00 STEAM-ENGINES AND BOILERS. Berry's Temperature-entropy Diagram lamo, i 25 Carnot's Reflections on the Motive Power of Heat. (Thurston.) i2mo, i 50 Dawson's "Engineering" and Electric Traction Pocket-book. . i6mo, mor., 5 00 Ford's Boiler Making for Boiler Makers i8mo, i 00 Goss's Locomotive Sparks 8vo, 2 00 Hemenway's Indicator Practice and Steam-engine Economy i2mo, 2 00 Button's Mechanical Engineering of Power Plants 8vo, 5 00 Heat and Heat-engines 8vo, 5 00 Kent's Steam boiler Economy 8vo, 4 00 Kneass's Practice and Theory of the Injector 8vo, i 50 MacCord's Slide-valves 8vo, 2 00 Meyer's Modern Locomotive Construction 4to, 10 00 Peabody's Manual of the Steam-engine Indicator i2mo. i 50 Tables of the Properties of Saturated Steam and Other Vapors 8vo, i oo Thermodynamics of the Steam-engine and Other Heat-engines 8vo, 5 00 Valve-gears for Steam-engines 8vo, 2 50 Peabody and Miller's Steam-boilers 8vo, 4 00 Pray's Twenty Years with the Indicator Large 8vo, 2 50 Pupin's Thermodynamics of Reversible Cycles in Gases and Saturated Vapors. (Osterberg.) i2mo, i 25 Reagan's Locomotives: Simple Compound, and Electric i2mo, 2 50 Rontgen's Principles of Thermodynamics. (Du Bois.) 8vo, 5 00 Sinclair's Locomotive Engine Running and Management i2mo, 2 00 Smart's Handbook of Engineering Laboratory Practice i2mo, 2 50 Snow's Steam-boiler Practice 8vo, 3 00 Spangler's Valve-gears Svo, 2 50 Notes on Thermodynamics i2mo, 1 00 Spangler, Greene, and Marshall's Elements of Steam-engineering Svo, 3 00 Thurston's Handy Tables Svo, i 50 Manual of the Steam-engine 2 vols., Svo, 10 00 Part I. History, Structure, and Theory Svo, 6 00 Part II. Design, Construction, and Operation Svo, 6 00 Handbook of Engine and Boiler Trials, and the Use of the Indicator and the Prony Brake Svo, 5 00 Stationary Steam-engines Svo, 2 50 Steam-boiler Explosions in Theory and in Practice i2mo, i 50 Manual of Steam-boilers, their Designs, Construction, and Operation Svo, 5 00 Weisbach's Heat, Steam, and Steam-engines. (Du Bois.) Svo, 5 00 Whitham's Steam-engine Design Svo, 5 00 Wilson's Treatise on Steam-boilers. (Flather.) l6mo, 2 50 Wood's Thermodynamics, Heat Motors, and Refrigerating Machines. . .Svo, 4 00 MECHANICS AND MACHINERY. Barr's Kinematics of Machinery Svo, 2 50 * Bovey's Strength of Materials and Theory of Structures Svo, 7 50 Chase's The Art of Pattern-making i2mo, 2 50 Church's Mechanics of Engineering Svo, 6 00 Notes and Examples in Mechanics Svo, 2 00 Compton's First Lessons in Metal-working i2mo, i 50 Compton and De Groodt's The Speed Lathe i2mo. i 50 14 Cromwell's Treatise on Toothed Gearing i2mo, i 50 Treatise on Belts and Pulleys i2mo, " 50 Dana's Text-book of Elementary Mechanics for Colleges and Schools. .i2mo, i 50 Dingey's Machinery Pattern Making i2mo, 2 00 Dredge's Record of the Transportation Exhibits Building of the World's Columbian Exposition of 1893 4to half morocco, 5 00 Du Bois's Elementary Principles of Mechanics: Vol. I. Kinematics 8vo, 3 50 Vol. II. Statics 8vo, 4 00 Mechanics of Engineering. Vol. I Small 4to, 7 50 Vol. II Small 4to, 10 00 Durley's Kinematics of Machines 8vo, 4 00 Fitzgerald's Boston Machinist i6mo, 1 00 Flather's Dynamometers, and the Measurement of Power i2mo, 3 00 , Rope Driving i2mo, 2 00 Goss's Locomotive Sparks 8vo, 2 00 * Greene's Structural Mechanics 8vo, 2 50 Hall's Car Lubrication i2mo, 1 00 Holly's Art of Saw Filing iSmo, 75 James's Kineruiitics of a Point and the Rational Mechanics of a Particle. Small 8vo, 2 00 * Johnson's (W. W.) Theoretical Mechanics i2mo, 3 00 Johnson's (L. J.) Statics by Graphic and Algebraic Methods 8vo, 2 00 Jones's Machine Design: Part I. Kinematics of Machinery Svo, i 50 Part II. Form, Strength, and Proportions of Parts Svo, 3 00 Kerr's Power and Power Transmission Svo, 2 00 Lanza's Applied Mechanics Svo, 7 50 Leonard's Machine Shop, Tools, and Methods Svo, 4 00 * Lorenz's Modern Refrigerating Machinery. (Pope, Haven, and Dean.). Svo, 4 00 MacCord's Kinematics; or. Practical Mechanism Svo, 5 00 Velocity Diagrams Svo, i 50 Maurer's Technical Mechanics Svo, 4 00 Merriman's Mechanics of Materials Svo, 5 00 * Elements of Mechanics i2mo, i 00 * Michie's Elements of Analytical Mechanics Svo, 4 00 Reagan's Locomotives: Simple, Compound, and Electric i2mo, 2 50 Reid's Course in Mechanical Drawing Svo, 2 00 Text-book of Mechanical Drawing and Elementary Machine Design. Svo, 3 00 Richards's Compressed Air i2mo, i 50 Robinson's Principles of Mechanism Svo, 3 00 Ryan, Norris, and Hoxie's Electrical Machinery. VoL I Svo, 2 50 Schwamb and Merrill's Elements of Mechanism Svo, 3 00 Sinclair's Locomotive-engine Running and Management i2mo, 2 00 Smith's (O.) Press-working of Metals Svo, 3 00 Smith's (A. W.) Materials of Machines i2mo, i 00 Smith (A. W.) and Marx's Machine Design Svo, 3 00 Spangler, Greene, and Marshall's Elements of Steam-engineering Svo, 3 00 Thurston's Treatise on Friction and Lost Work in Machinery and Mill Work Svo, 3 00 Animal as a Machine and Prime Motor, and the Laws of Energetics. i2mo, I 00 Warren's Elements of Machine Construction and Drawing Svo, 7 50 Weisbach's Kinematics and Power of Transmission. (Herrmann — Klein.). Svo, 5 00 Machinery of Transmission and Governors. (Herrmann — Klein.). Svo, 5 00 Wood's Elements of Analytical Mechanics Svo, 3 00 Principles of Elementary Mechanics i2mo, i 25 Turbines Svo, 2 50 The World's Columbian Exposition of 1893 4to, i 00 15 METALLURGY. Egleston's Metallurgy of Silver, Gold, and Mercury: Vol. I. Silver 8vo, 7 50 Vol. II. Gold and Mercury 8vo, 7 50 ** lias's Lead-smelting. (Postage p cents additional.) i2mo, 2 50 Keep's Cast Iron 8vo, 2 50 Kunhardt's Practice of Ore Dressing in Europe Svo, i 50 Le Chatelier's High-temperature Measurements. (Boudouard — Burgess.)i2mc. 3 00 Metcalf's Steel. A Manual for Steel-users i2mo, 2 00 Minet's Production of Aluminum and its Industrial Use. (Waldo.)... .i2mo, 2 50 Robine and Lenglen's Cyanide Industry. (Le Clerc.) 8vo, Smith's Materials of Machines i2mo, i 00 Thurston's Materials of Engineering. In Three Parts Svo, 8 00 Part II. Iron and Steel Svo, 3 50 Part III. A Treatise on Brasses, Bronzes, and Other Alloys and their Constituents Svo, 2 50 Ulke's Modern Electrolytic Copper Refining Svo, 3 00 MINERALOGY. Barringer's Description of Minerals of Commercial Value. Oblong, morocco, 2 50 Boyd's Resources of Southwest Virginia Svo, 3 00 Map of Southwest Virignia Pocket-book form. 2 00 Brush's Manual of Determinative Mineralogy. (Penfield.) Svo, 4 00 Chester's Catalogue of Minerals Svo, paper, i 00 Cloth, 1 25 Dictionary of the Names of Minerals Svo, 3 50 Dana's System of Mineralogy Large Svo, half leather, 12 50 First Appendix to Dana's New " System of Mineralogy." Large Svo, i 00 Text-book of Mineralogy Svo, 4 00 Minerals and How to Study Them i2mo. 1 50 Catalogue of American Localities of Minerals Large Svo, i 00 Manual of Mineralogy and Petrography i2mo, 2 00 Douglas's Untechnical Addresses on Technical Subjects i2mo, i 00 Eakle's Mineral Tables Svo, i 25 Egleston's Catalogue of Minerals and Synonyms Svo, 2 50 Hussak's The Determination of Rock-forming Minerals. (Smith.). Small Svo, 2 00 Merrill's Non-metallic Minerals: Their Occurrence and Uses Svo, 4 00 * Penfield's Notes on Determinative Mineralogy and Record of Mineral Tests. Svo, paper, 50 Rosenbusch's Microscopical Physiography of the Rock-making Minerals. (Iddings.) Svo, 5 00 * Tillman's Text-book of Important Minerals and Rocks Svo, 2 00 MINING. Beard's Ventilation of Mines i2mo, 2 50 Boyd's Resources of Southwest Virginia .» Svo, 3 00 Map of Southwest Virginia Pocket-book form 2 00 Douglas's Untechnical Addresses on Technical Subjects i2mo, i 00 * Drinker's Tunneling, Explosive Compounds, and Rock Drills. .4to,hf. mor., 25 00 Eissler's Modern High Explosives Svo 4 00 16 3 oo 4 00 I 50 2 50 I oo 3 SO 3 00 7 50 4 00 Fowler's Sewage Works Analyses i2mo, 2 00 Goodyear's Coal-mines of the Western Coast of the United States i2mo, 2 50 Ihlseng's Manual of Mining 8vo, 5 00 ** lles's Lead-smelting. (Postage gc. additional.) i2mo, 2 50 Kunhardt's Practice of Ore Dressing in Europe 8vo, i 50 O'Driscoll's Notes on the treatment of Gold Ores 8vo, 2 00 Robine and Lenglen's Cyanide Industry. (Le Clerc.) 8vo, * Walke's Lectures on Explosives 8vo, 4 00 Wilson's Cyanide Processes i2mo, i 50 Chlorination Process i2mo, i 50 Hydraulic and Placer Mining i2mo, 2 00 Treatise on Practical and Theoretical Mine Ventilation i2mo, i 25 SANITARY SCIENCE. Bashore's Sanitation of a Country House i2mo, Folwell's Sewerage. (Designing, Construction, and Maintenance.) 8vo, Water-supply Engineering 8vo, Fuertes's Water and Public Health i2mo, Water-filtration Works i2mo, Gerhard's Guide to Sanitary House-inspection i6mo, Goodrich's Economic Disposal of Town's Refuse Demy 8vo, Hazen's Filtration of Public Water-supplies 8vo, Leach's The Inspection and Analysis of Food with Special Reference to State Control 8vo, Mason's Water-supply. (Considered principally from a Sanitary Standpoint) 8vo, Examination of Water. (Chemical and BacteriologicaL) i2mo, i 25 Ogden's Sewer Design i2mo, 2 00 Prescott and Winslow's Elements of Water Bacteriology, with Special Refer- ence to Sanitary Water Analysis i2nio, i 25 * Price's Handbook on Sanitation i2mo, j 50 Richards's Cost of Food. A Study in bietaries i2mo, i 00 Cost of Living as Modified by Sanitary Science i2mo, i 00 Richards and Woodman'-s Air. Water, and Food from a Sanitary Stand- point 8vo, 2 00 * Richards and Williams's The Dietary Computer 8vo, i 50 Rideal's Sewage and Bacterial Purification of Sewage 8vo, 3 50 Turneaure and Russell's Public Water-supplies 8vo, 5 00 Von Behring's Suppression of Tuberculosis. (Bolduan.) i2mo, i 00 Whipple's Microscopy of Drinking-water 8vo, 3 50 Winton's Microscopy of Vegetable Foods 8vo, 7 50 Woodhull's Notes on Military Hygiene i6mo, i 50 MISCELLANEOUS. De Fursac's Manual of Psychiatry. (Rosanofif and Collins.). . . .Large i2mo, 2 50 Emmons's Geological Guide-book of the Rocky Mountain Exctxrsion of the International Congress of Geologists Large 8vo, i 50 Ferrel's Popular Treatise on the Winds 8vo. 4 00 Haines's American Railway Management i2mo, 2 50 Mott's Fallacy of the Present Theory of Sound i6mo, i 00 Ricketts's History of Rensselaer Polytechnic Institute, 1824-1894. .Small 8vo, 3 00 Rostoski's.Serum Diagnosis. (Bolduan.) i2mo, i 00 Rotherham's Emphasized New Testament Large 8vo, 2 00 17 Steel's Treatise on the Diseases of the Dog 8vo, 3 50 The World's Columbian Exposition of 1893 4to, i 00 Von Behring's Suppression ot Tuberculosis. (Bolduan.) iimo, i 00 Winslow's Elements of Applied Microscopy i2mo, i 50 Worcester and Atkinson. Small Hospitals, Establishment and Maintenance; Suggestions for Hospital Architecture: Plans for Small Hospital. i2mo, 1 25 HEBREW AND CHALDEE TEXT-BOOKS. Green's Elementary Hebrew Grammar i2mo, 1 25 Hebrew Chrestomathy 8vo, 2 00 Gesenius's Hebrew and Chaldee Lexicon to the Old Testament Scriptures. (Tregelles.) Small 4to, half morocco, 5 00 Letteris's Hebrew Bible Sto, 2 25 18 ^Ct /<3-i?-^ t-^M LOAN DEPT. This book is d« o» *e .js, d„. "SSin'S; i „a >he date .o «h^* «« '< -^'^ ■" -,:! r., % ??■ -i/ LOANl ^ ?5i — >