556 



AX ESSAY OX CyCLOIDAL CURVES. 



lishman on his travels^ who visits a country 

 by drivingAvith all possible speed from place 

 to place by night, and refreshing his fatigues 

 in the day time, by lounging half asleep at 

 his hotel. Undoubtedly there are some coun- 

 tries through which one may reasonably wish 

 to travel by night, and uiidoubiodly there are 

 Bome cases where algebraical symbols areniore 

 convenient than geometrical ones : but 

 ■when we see an author exerting all his inge- 

 nuity in order to avoid every idea that has 

 the least tincture of geometry,' when he 

 obliges us to toil through immense volumes 

 filled with all manner of literal characters, 

 without a single diagram to diversity the 

 prospect, we may observe with the less sur- 

 prise, that such an "author appears to be con- 

 fused in his conception of the most elemen- 

 tary doctrines, and that he fancies he has 

 made an improvement of consequence, when, 

 in fact, he is only viewing an old object in a 

 new disguise. It happens frequently in the 

 description of curves, and in the solution of 

 problems, that the geometrical construction 

 is very simple and easy, while it almost ex- 

 ceeds the powers of calculus to express the 

 curve or the locus of the- equation in a man- 

 ner strictly algebraical :, and, indeed, the 

 astonishing advances that were n;ade, in a 

 comparatively short time, by Euclid, by Apol- 

 lonius, and above all, by Archimedes, are 

 sufficient to prove, that the method of repre- 

 sentation which they employed could not be 

 very limited in its application : and the pre- 

 cision and elegance, with which the method 

 of geometrical fluxions is treated by Newton 

 and Maclaurin, form a strong contrast to the 

 tedious affectation of abstraction an(l obscu- 

 rity which unfortunately pervades the writings 

 of many great mathematicians of a later date. 

 It would be of inestimable advantage to the 



progress of all the sciences, if some diligent 

 and judicious collector would undertake to 

 compile a complete system of mathematics ; 

 not as an elementary treatise, nor as a mere 

 index of reference, but to contain every pro- 

 position, with a concise demonstration, that 

 has ever yet been communicated to the pub- 

 lic. Until this is done, nothing is left but for 

 every individual, who is curious in the search 

 of geometrical knowledge, to look over all 

 the mathematical authors, and all the literary 

 memoirs, of the last and present centuries.: 

 for without this, he may very easily fancy he 

 has made discoveries, when the same facts 

 have been known and forgotten long before 

 he existed. An instance of this has lately 

 occurred to ayounggentlemanin Edinburgh, 

 a man who certainly promises, in the course 

 of time, to add considerably to our know- 

 ledge ol' the laws of nature. The tractory, 

 tractrix, or equitangential curve, was first 

 described by Huygens, and afterwards more 

 fully by Mr. Bomie, (Mem. Acad. ]712,)and 

 by Mr. Perks. (Ph. tr. XIX. n. 345, Abr. 

 IV. 456.) Bomie and Perks have shown 

 many remarkable properties belonging to it ; 

 and one in particular, which may be briefly 

 demonstrated, that it is the involute of the 

 catenaria: for since the equation of the ca- 

 tenaria is zz r: 2«x + xx, wq have zz — ax 

 -f XX, and x : z :: 2 : a + x, therefore the 

 vertex of the right angled triangle, •bf which 

 the base is the evolved radius, and the hypo- 

 tenuse a line paralhel to the axis of the curve, 

 describes a right hue ; and the perpendicu- 

 lar of this triangle is always = a, and is the 

 constant tangent of the curve described by 

 the evolution. Cotes has also, in his Logo- 

 metria, investigated the properties of the 

 tractrix of the circle. Bernoulli observed 

 in 1730, that the tractrix was one of the 



