PRO.TF.rT 



PROJECTION OK MATHKMATICAL DIAGRAMS. 71 



an bouncUriM of * figure on the second, which u the projection * o 

 OM first figure. 



It pr*-h*p not usual to nuke so wide a definition of projection it 

 tin 1 1 (iDoe the only cue* which are commonly considered are those 

 in which the projecting line* are all straight, and either parallel to one 

 another, at in the orthographic projection, or all paaaing through the 

 same point, a* in common perspective. But such a conception o 

 projection U necenary : in Meroator's projection, for example, [MAI- 

 the point* of a *phere are projected on a uircumncribing cylinder, nu 

 by straight line* panting through a point, but either by straight line* 

 dupoand according to a complicated law, or elie by curved. If a 

 relation between any point ami it* projection be given, *o that either 

 can be found from the other, the passage from one to the other may be 

 made either on a utraight line or on an infinite vnrioty of curve* ; bin 

 it may happen that the law which the disposition of the projecting 

 straight line* follow* may be of a more difficult character than thai 

 which would be required If a curve, not in itaelf to simple a* a straight 

 line, were mib*titut<-d. 



When the foundations of plane geometry were fixed, and the first 

 principle* of solid geometry were superseded, it was natural that the 

 very simple idea of the perspective projection should excite atU-mi. n 

 In a country in which the Bret principle* at least of drawing were 

 practically known, the following problem must have suggested itself to 

 the geometer* : If through a given point lines be drawn through all 

 .the point* of the boundary of a plane figure, until they are stopped by 

 another plane, required the figure traced out upon the second plane. 

 'A straight Une was known thus to give a straight line : a morn. m'.-. 

 consideration of the circle, the only other line wen considered, would 

 show that a projection of a circle and a plane section of a cone are the 

 ame thing*. Hence probably the first idea of a conic section ; and 

 thus, if the conjecture be correct, the attention was turned from that 

 point, which would, if properly kept in view, have led to the theory o) 

 projections in place of one isolated branch of it. The properties of the 

 'come sections, as deduoed in the ancient manner from the cone, are 

 neither so general nor so easy a* they might be made ; and it may be 

 confidently expected, considering the progress which the doctrine of 

 projections has made of late years, that the method of considering the 

 ellipse, hyperbola, and parabola as projections of the circle, will 

 become established in elementary teaching, in preference to the 

 detached geometrical and algebraical methods now in use. 



We have already spoken of the geometry of projections [GEOMETRY] : 

 unfortunately there is no elementary work which give* a general view 

 of its first principles supported by sufficient application ; and until 

 uch a work shall aj>i>ear, the student must search for himself the 

 writings of Mongc, Comfit, Chaales, Poncelet, Ac. The ' History of 

 Geometry,' by M. Charies, referred to in the article cited, will furnish 

 many more references ; and the ' Proprie'te* Projectives des Figures,' 

 by M. Poncelet, is jwrhap* the work in which the student may most 

 easily make an advantageous beginning of the subject Much has been 

 done of late years to introduce some element* into works on the conic 

 ections : but projection has not yet been properly treated as a 

 distinct subject. 



The basis of the theory of projections must be the investigation of 

 properties which, being true of a figure, are therefore true of its 

 projections. Some of these are evident enough : thus the projection 

 Df an intersection of two lines is the intersection of their projections ; if 

 two curves touch one another, their projections touch at a point which 

 U the projection of the point of contact. But the following property, 

 which is projective, that is, true of the projections of every figure of 

 which it is true in the first instance, will give a good idea of the 

 facility with which certain properties of the conic sections may be 

 Jeduced from the circle. 



usual way. We might say, in more geometrical language, let there be 

 any number of ratios which, compounded together, give either a ratio 

 of equality or a given ratio. Two simple conditions being fulfilled, 

 this property will lie as true of the projections as it is of the figure 

 These conditions are, first, that every initial and terminal 

 etter shall occur the same number of times on both sides of the 

 equation; secondly, that for every line on the first side, there shall be 

 a distinct line on the other side, which is in the same straight line. 

 For example (the reader may draw the diagram for himself), let each 

 of the rides of the triangle ABC cut one circle, namely, AB in r 

 and Q, BC in K and R, c A in T ami v ; the order of the point* being 

 AI-QBBSCTVA. Then by the properties of the circle, it is easily 

 Been that * 



A V. A T. C S. C II. B Q. B r = A P. A <J. B B. B 8. C T. C V. 



;In this equation, A, n, and c occur twice on each rido, and each of 



P, Q, R, B, T, v.f.nce. Moreover, nut of AB there are two segments, 



A r and A Q., .,11 the first side, and na many, no. and B r, on the second; 



the same of BC and CA. He then who is acquainted with (!.. 



theory of projections, immediately knows that this property is true of 



The projection of * Hat or flipire hu seldom any other name ; in tio- 

 jiomj however, the projection of plsntt'n ill-Unco from too ion on the plmnc of 

 the ecliptic U sometime* called the curtate distance. 



any projection of a circle, or of any conic aection : but h,- would be an 

 energetic algebraist who should attempt ! prove this (or still more 

 the equally demonstrable similar property in the case of a polygon ot 

 n sides) by the common algebraic methods, 



The proof of the preceding general prtyc(i property i* nc.t difficult. 

 Take a point o outside the plane for the centre of projection, and let 

 o A=O, OB = 4, etc. ; moreover let the angle made by a and b be 

 (06). Let A', B*, Ac, be the projection* of A, B, Ac., letOA' = o', 

 o B' = V, Ac., and let (a' 6') be the angle of o' and !>', which is = (a b). 

 Moreover let [A B] mean the perpendicular let fall from o upon A B, Ac. 

 It i* then easily proved that 



Substitute them values in the equation, and it will be readily seen 

 that th ,.f the Uv i above named amount* to aO 



the t]u;uititicH except the sines of the angles being eliminable by <l: 

 There remains then 



sin !). Ac. = 



or sin (a r'). sin (a' /'). &c. = 



i (07). Ac. 



Bill ('/'). Mil ('</). &C. 



In this write a', V, [A' v'], Ac., where there were previously a, b, [A v]' 



Ac., which will amount (by the conditions) to multiply"! 



liy the same quantities : there will then remain an equation which is 



obviously 



A'V'. A'T-. AC. = A'P / . A'Q'. Ac., 



and in the same way any other case may be proved. 



PROJECTION OF MATHEMATICAL DIAGRAMS. The dia- 

 grams by which mathematical students (and even writers) represent 

 their solid figures are generally so imperfect, that it maybe worth 

 while to explain how, in all cases of sufficient importance, a good 

 drawing may be made with very little trouble. The demount : 

 may be found in the ' Cambridge Mathematical Journal,' N.I. S p. n-j. 

 The projection is supposed to be the ORTHO^H. \riu< . in whirh the 

 eye is at an infinite distance, and all parallels ore projected into 

 parallels, Ac. 



Let o \, T, o z, bo the intended projections of the three axes of 

 co-ordinates, the dark lines being supposed to belong to tint quarter 

 of space in which lies a line drawn to the eye from the origin o. 

 of the angles Y o z, z o x, X o Y, is then greater than a right angle. The 

 following table contains numbers sufficiently near for the purpose, 

 proportional to the square rooU of the Bines of twice the angle* written 

 in the opposite columns. 



90-180 000 

 132 



91-179187 



229 



92-178264 



295 



93-177323 



349 



94-176 373 

 396 

 417 

 437 

 456 

 474 

 97-17 



J509 



98-172525 

 -'541 



105-165707 



- 7is 



- ,738 

 107-163748 



7. -.7 



95-175 

 96-174 



108-162767 



77 



109-161 785 







110-lCu >oi 



-11 



111-159818 



99171 



[571 



100-170-585 



.599 



101-169,612 



625 



102-16S638 



'060 



103-167662 







104-106 685 



112-168834 

 841 

 848 

 855 

 156868 



115-155875 



8S2 



116-154888 



894 



117-153 UOO 



906 



118-1. 







119-1. 



