456 ' PHILOSOPHICAL TRANSACTIONS. [aNNO lOgQ. 



determines the triangle alc equal to the portion ade. Which being admitted, 

 we may thus divide the lunula in any given proportion. If we divide ab at l, 

 in such given proportion ; cl will in the same proportion, because of the com- 

 mon altitude, divide the triangle acb, which is equal to the whole lunula. And 

 LE, erected at right angles on alb, will determine the point e ; from whence if 

 we draw the straight line ec, this will at de, divide the lunula in the same pro- 

 portion. 



Mr. Perks, on edc, drawing the perpendicular af, fig. 5, determines the 

 semiquadrate afe, equal to the proposed portion ade. Which semiquadrate is 

 a like figure, and alike situated to ae, as is acb to ab. And therefore, because 

 like figures are in the duplicate proportion of their respective sides, if we so in- 

 scribe AE, as that the square of ae be to the square of ab in such given propor- 

 tion, the lunula will at de be divided as is required. And this will hold, if duly 

 applied, according as the different cases may require, though e be taken, in the 

 continuation of the semicircle, beyond b. For still like figures will be in dupli- 

 cate proportion of their respective sides, and ce = cd ± de. And the same is 

 yet improveable much further. John Wallis. 



Answer to the Animadversions concerning the Catenary. By Dr. David Gregory. 

 Translated from the Latin. N° 259, p. 419. 



What has been objected by an anonymous author, in the Leipsic Acts of 

 Feb. 1699, in his animadversions on my demonstrations concerning the caten- 

 ary, is this : that I have undertaken to demonstrate, after my manner, a matter 

 found out and published by others seven years ago. This is true, and I cannot 

 find any thing in it that is blame worthy. Those great men Huygens, 

 Leibnitz, and Bernouilli, have discovered and communicated many properties of 

 the catenaria, but without demonstration. I have contrived demonstrations, 

 which was tiie thing I undertook to do. 



But was this matter (that is, the nature and primary properties of the 

 catenaria) all found out and published by others .-' Surely that property of the 

 catenaria, in Cor. 6, Prop. 2, was not at all mentioned by others, before the 

 publication of these demonstrations ; although, if I am not mistaken, it may 

 be reckoned among its primary properties, and is the most useful of all, and 

 most easily reduced to the common purposes of life. From all ages architects 

 have made use of arches in public buildings, as well for strength as beauty. Yet 

 what was the true geometrical figure of an arch was not known before my de- 

 monstrations came out. 



The first he finds fault with is, that I affirm some things are plain from me- 

 chanics, which he thinks should have been explained and applied more distinctly. 



