NA TURE 



[December 29, 1904 



knowledge of a large assortment of theorems is necessary ; 

 but the practical value of the study to students who are 

 neither engineers nor architects is another matter. 



There is, however, another kind of mathematical drawing 

 which does not fall under any of these heads, and which 

 consists in the invention of graphic solutions of equations 

 which can be solved with great difficulty, if at all, by the 

 stock processes of accurate mathematics. This branch is 

 at once the most useful and the most vague ; it is impossible 

 to lay down its principles in systematic order — it must be 

 learnt by abundant exemplification. 



The ordinary academic problems of statics and hydro- 

 statics furnish many e.\amples of this subject, but only a 

 few of these can be noticed here. 



If AB and BC are two ladders freely jointed together at 

 B, of different weights and lengths, placed with the ends 

 A and C resting on a rough horizontal plane, A being pre- 

 vented from moving while C is drawn out along the plane, 

 the inclinations, 9, 0, of AB and BC to the ground when 

 the limiting position is reached are determined from two 

 equations of the forms 



a sin 6 — h sin ^ =o ; m tan B+n tan (^ =fe, 

 where a, b, m, n, k are all given quantities. The graphic 

 solution of these equations is effected with great ease thus : — 

 draw a line OH equal to m, and produce OH to O' so that 

 HO'=i!; at H draw HC perpendicular to OO' and equal 

 to k ; through O draw anv line OQ meeting HC in Q ; take 

 a point R in CH such that CR = HQ, and draw O'R; then 

 the point, P, of intersection of OQ and O'R is a point on 

 the locus represented bv the second of the above equations, 

 the angles $, <t> being POO' and PO'O. These points, P, 

 are therefore constructed with great ease and rapidity. ."Mso 

 the locus represented by the first equation is a circle having 

 for diameter the line joining the points which divide OO' 

 internally and externally in the ratio a : b, and the points 

 of intersection of these two loci give the required values of 

 e and iji. 



The following problem leads to precisely the same equa- 

 tions as the above : — rays of light emanate from a fixed 

 point P in one medium separated by a plane surface from a 

 second medium ; find the ray proceeding from P which will 

 be refracted to a given point, O, in the second medium. 



Again, the fact that when a uniform chain hangs with 

 free extremities over two fixed supports of equal heights 

 there are either two figures of equilibrium or none results 

 from the solution of an equation of the form xe<'l^ = k, which 

 is effected by drawing the curve y = ei and the right line 

 jr = fe«/a, and then it is at once seen that there are either 

 two points of intersection or none. 



When a heavy wire rope has its ends fixed at two points 

 in the same horizontal line, and a load is suspended from 

 the lowest point of the rope, the rope forms parts of two 

 distinct catenaries, and the determination of these curves 

 leads to an equation of the form 



("/'^[{x"- + a^)'' + aM^- + *-)♦ + ''I 

 in which x alone is unknown. The tracing of the curve 

 obtained by putting y equal to the right-hand side of this 

 equation is quickly effected by means of two fixed circles 

 and the drawing of right lines. 



The figure of equilibrium of a revolving self-attracting 

 liquid spheroid gives an equation which is a particular case 

 of x(a+hx-)!(c + x') = tSLn-^x, and this is best solved by the 

 tracing of two curves. If we put y equal to the left-hand 

 side we have a curve of the third degree the geometrical 

 construction of which is exceedingly simple, and requires 

 only a fixed circle and right lines. 



Whenever a problem involves two unknown angles in two 

 equations one of which is of the form m cos 8+n cos <!> = €, 

 where m. n, c are given, all angles satisfying this equation 

 can be represented as the base angles of a triangle the base 

 of which, AB, is fixed, and the vertex of which describes 

 what may be called a quasi-magnetic curve, the geometrical 

 construction of which is this : take any two fixed points, 

 A, B ; about A as centre, with radius m.AB/c describe .t 

 circle; about B describe a circle with radius ti.AB'c; draw 

 any line perpendicular to AB meeting these circles in Q and 

 R respectively ; then the lines AQ and BR intersect in a 

 point on the required curve. When m = n we have the 

 common magnetic curve the construction of which is not 

 nearly so well known as it should be. 



NO. 1835, VOL. 71] 



The solutions of the above examples have all been of a 

 purely geometrical kind, and have not involved the plotting 

 of points by coordinates arithmetically calculated. There 

 are other problems of a slightly different kind, still in- 

 dependent of plotting, but involving trial ; the value of a 

 certain unknown quantity which has to satisfy a certain 

 geometrical condition is found by trial to do so very nearly 

 if not completely. In all such cases Taylor's theorem 

 furnishes a still closer value than the observed one, and 

 completes the solution with all desirable accuracy. 



For example, many problems lead to the equation 

 a sin 2(9 — a) = b sin B for an unknown angle 6, the other 

 quantities being all given. This can be solved by two 

 circles thus : — draw a line AB equal to h, and on it as 

 diameter describe a circle the centre of which is C ; draw 

 AD making the angle BAD=o and cutting the circle in D ; 

 draw CD and produce it to E so that CE = a, and on CE 

 as diameter describe a circle. \ow find on the circum- 

 ference of the first circle a point P such that if CP meets 

 the second circle in Q we have BP = EQ. This is done with 

 great accuracy by the eye, and Taylor's theorem will im- 

 prove the solution. 



An equation which can be solved also very easily by trial 

 is a sin^9 = 6 cot 6, which may be taken in the form 

 a sin' 9 = ft cos d, and a graphic solution suitable to each 

 form is easily found. 



Finally, we may notice equations of the form 



tan x = axl(c — x^), 



which we obtain from Bessel functions in certain problems 

 relating to vibrations. Such an equation is easily solved 

 by the intersections of the curve y = cot x with the hyperbola 

 y = (c — x')lax, and the construction of the hyperbola belongs 

 to the most simple case of this curve, viz. given one point 

 on the curve and the asymptotes. \s compared with the 

 graphic solution of equations given by physical problems, 

 the graphic solution of algebraic equations is unimportant, 

 though not devoid of interest, because Horner is always 

 available for numerical cases. 



Prof. Gibson gives many examples of the solutions of 

 quadratics and of cubics by graphic methods ; but as re- 

 gards quadratics it must be confessed that there is no 

 utility in the process, and too much space is usually devoted 

 to it. For cubics in general he gives a graphic solution 

 and an interesting discussion. In a second edition of his 

 book he might treat the biquadratic similarly, because its 

 graphic solution can be easily effected by means of a circle 

 and a parabola, or by means of a right line and a curve 

 easily derived from a parabola. Many curves occurring in 

 phvsics are dealt with in the book — such as isothermals 

 and adiabatics ; there is also a useful discussion of Fourier's 

 theorem, and a treatment of the curves belonging to vibra- 

 tions, damped as well as undamped. The graphic method 

 is also applied to the solution of some of the simpler mixed 

 trigonometric and algebraic equations, and the book con- 

 cludes with a chapter on the properties of conic sections. 

 George M. Mixchin. 



CENTRAL AMERICAN MAMMALS.' 

 T^HREE years ago the author of these volumes published, 

 in the same serial, a valuable synopsis of the mammals 

 of North America and the adjacent seas. In the present 

 larger work he has taken in hand the mammals of the tract 

 generally known in this country as Central America, but 

 on the other side of the .Atlantic termed, at any rate by 

 zoologists. Middle America, together w-ith those of the West 

 Indian islands. The greater bulk of the present work is 

 accounted for, not so much by the greater number of species 

 (690 against 606) as by the increased elaboration of 

 the mode of treatment, the addition of diagnostic " keys " 

 to the various genera, and by a fuller account of the habits 

 of many species, the latter feature rendering these volumes, 

 proportionately more valuable to the naturalist, and at the 

 ■;ame time of more general interest. The illustrations, 

 too, are more numerous, comprising, besides crania, figures 

 of the external form of a considerable number of species. 



1 "The Land and -Sea Mamm.-ils o' MidHlc America and the Wesi 

 Indies." By D. C. Elliof. /VWrf ColumHan Miiuum P::Hicalions, 

 Zoohgical Series, vol. iv., part^ 1. and il , pp. xxi+850. illusITaled. 



