June -5, 1890] 



NATURE 



139 



NEWTON'S INFLUENCE ON MODERN 

 GEOMETRY. 



TN the appendix to his "Arithmetica Universalis" Newton states 

 -'■ that a study of the ancient philofopheis had led him to the 

 inevitable conclusion that those early pioneers of science had 

 introduced geometry in order to escape from needlessly long and 

 laborious calculations. So, too, the author of the " Principia" 

 had a predilection for graphic as distinguished from analytic 

 methods. Indeed, anyone who has perused that great work will 

 readily endorse the truth of this assertion. Yet Newton was 

 born some forty years after the death of Viete and only eight 

 before that of Rene Descartes, whose writings gave such a 

 wondrous impulse to analytical studies. 



During the closing period of the seventeenth, and nearly the 

 whole of the eighteenth century, analysis reigned supreme; 

 whilst graphic methods languished from the wilful neglect, nay 

 even undisguised contempt, of the new philosophers. But at 

 length men grew weary of abstract thought, and, as was quite 

 natural after an undue pursuit of one branch of science to the 



exclusion of all others, a strong reactionary current in favour of 

 concrete geometrical studies supervened. Then, as now, the 

 question of the respective merits of the two methods gave rise 

 to serious, not to say heated, controversy. But why sane people 

 should quarrel and then fall out over a purely mathematical 

 difference of this sort is quite as incomprehensible to a sober- 

 minded critic as the passionate resistance shown to the postal 

 reforms of Sir Rowland Hill was to the placid and imperturbable 

 mind of Lord Melbourne. 



The general weariness of the scientific mind, brought on by 

 an excess of analytical work, prepared the way for a great 

 revival of the graphic culte. Carnot, following to a certain 

 extent the previous example of Simpson, courageously resolved 

 to continue the work of Pascal, Newton, and Desargues. In 

 consideration of his treatises on projective geometry and the 

 theory of transversals, Carnot has a definite claim to be deemed 

 the leader of this modern insurrection against the excessive use 

 of analysis. Contemporary with him we find Monge, one of 

 whose pupils, Poncelet, may be justly termed the author far 

 excellence of modern methods. Since Poncelet's time the further 



development of the system has been confined for the most part 

 to Germany and Switzerland, under the guidance of such leaders 

 as Stelner and Staudt. But, unfortunately, Staudt undertook 

 the arduous, if not impossible, task of expounding projective 

 geometry without the aid of diagrams, in regard to which 

 Hankel well remarks, "that such an attempt was possible only 

 in Germany, the land of scholastic methods and scientific 

 pedantry." 



Strange to say, Culmann, who was nothing if not a practical man 

 of science, presupposes a knowledge of Staudt's geometry in all 

 who would rightly understand his own epoch-making work on 

 graphic statics.^ Luckily, however, it is possible to understand 

 every line of Culmann without having read a single word of 

 Staudt. Now it is precisely the object of this paper to show 



' Published in the year 1864, not, as was recently stated in a contemporary, 

 in 1866. The date is of importance when discussing priority of discovery in 

 the matter of reciprocal figures ; for Maxwell's paper on the subject in the 

 Philosophical Magazine was also published in 1864. The question cannot, \ 

 however, be discussed in a footnote. j 



that, in some of its more salient features, this so-called Geometrie 

 der Lage is but a luxuriant offshoot of Newton's " Principia," in 

 illustration of which we will here proceed to prove how the 

 general method of constructing a conic, five points on which are 

 given, may be deduced from the similar proposition in Newton. 

 Further, in order to make the connection between Newton and 

 Staudt more apparent, it will be advisable first to give the 

 modern solution of the problem, and then show how the same 

 solution can be geometrically deduced from Newton's principle. 



Solution. — Take any two of the given five points, for 

 instance S and Sj (Fig. i), as centres of projection. Through a 

 third point, A, draw any two lines u and Mj. Then, from the 

 centre Sj, project the remaining two points B and D by rays 

 SjB and S^D intersecting line «i in the points b^ and (/j. 



Similarly, from the second centre S, project the same two 

 points B and D by rays intersecting the line u in b and d. Join 

 bb^ and dd-^, meeting in the centre of perspectivity, S^' of ^^ 

 lines tt and Wj. 



Then, to find a sixth point on the curve, draw any ray through 



NO. 1075, VOL. 42] 



