The first of these inferences was deduced

from the forty-seventh proposition in his first book. From this proposition it was evident that

if a right angle be drawn at the centre of a circle, the square of the side joining the ends of the radii is double the square of the radius; that is, ab square is double the square of ac. (see the

d

e

figure.) It was also evident that the square of the diametor ae is equal to four times the square of the radius ac (corollary to proposition 6, Supplement). Hence it was an obvious inference that the square of some line greater than ab and less than the diametor ae, would be triple of the square of the radius ac. The author was thereupon led to the attempt to find such a line; and, in the result, he found that the square of the line ad, being one of the sides of an equilateral triangle inscribed in the circle, is triple the square of the radius ac. (This is demonstrated in proposition 10, Supplement.)

Again, Euclid has not in any part of the Elements shown in what manner an angle may be trisected, much less has he given any general problem for dividing an angle into any number whatever of equal parts or angles. It appeared worthy

of the present author's pains to attempt the solution of both these problems; which he has accomplished in propositions 1, 23, and 38 of this Supplement; which he trusts will be considered no unimportant addition to this branch of science.

So, in the fourth book of the Elements which contains problems for inscribing polygons in circles, and describing polygons about circles, there is no general problem for constructing a polygon of any number of equal sides whatever; which appeared to the author a great desideratum, and which he has accordingly supplied in effect in proposition 38 of the Supplement.

The author has also considered it not superfluous to demonstrate in proposition 35, Supplement, that the squares of any two right lines are to each other in the duplicate ratio of the two right lines, there being no demonstration of that important property in all the Elements. By means of that proposition the nineteenth proposition of the sixth book of Euclid, viz. that similar triangles are to one another in the duplicate ratio of their homologous sides becomes turned into the following one, viz. that similar triangles are to one another as the squares of their homologous sides; as is demonstrated in the thirty-seventh proposi

tion in this work. That proposition would have saved Mr. Bridge and other authors on Mechanics much trouble in showing that the spaces described by a gravitating body may be represented by similar triangles of which two homologous sides represent the respective times during which the body gravitates. For it is ascertained by experiment and observation that the spaces described by a heavy body in different times are as the squares of the times. Therefore let abc, ade be two similar triangles, of which ab, ad are homologous sides; and let ab be equal to bd, and consequently ad the double of ab; and if the side ab represents one time d

b

a

e

of descent (for instance, one second); and if ad represents double that time (or two seconds), then will the triangle abc represent the space described in one second, and the similar triangle ade will represent the space described in two seconds.

For the spaces are as the squares of the homologous sides ab, ad, which squares are as the triangles abc, ade (Proposition 36, Supplement), wherefore ex equali, the spaces are as the triangles abc, ade, and may therefore be represented by them. Mr. Bridge, and other authors following him, by a laboured reasoning from approxima

tion, maintain this proposition, which admits of the above easy and direct demonstration.

Mr. Bridge, in demonstrating that the spaces may be represented by two similar right-angled triangles, assumes that the gravitating body receives equal increments of velocity in equal times (page 23, Bridge's Mechanics). But that property of the gravitating body is rather to be inferred from the properties of the similar triangles which represent the spaces described. For the triangles not only represent the spaces described, but the manner of their generation, and therefore the bases or sides bc and de represent the velocities acquired at the end of the two times ab and ad. For, whatever the velocities be, still they must be such as to generate spaces equal to the two similar triangles in the times ab, ad; and therefore these velocities must be to each other as the base bc is to the base de. And because the triangles are similar, their homologous sides are proportionals (6, Euclid 4), and therefore ad ab de bc; but ad is the double of ab; wherefore de is the double of be; that is, the velocity acquired at the end of two seconds is double the velocity acquired at the end of one second.

:

The author, in the 39th proposition in this work, has directly demonstrated that similar triangles are to each other as the rectangle of any two sides of one triangle is to the rectangle of the two homologous sides of the other triangle; which is a demonstration of Mr. Bridge's proposition (page 26) that the triangle abc is to the triangle ade, as the rectangle ab, bc is to the rectangle ad, de; which proposition that author does not demonstrate in the manner of Euclid.

Since in this work the propositions above mentioned are intended in some measure as the foundation of some of the mechanical operations respecting gravitation, the author cannot help noticing the inverted order of the reasoning of Mr. Bridge in his Mechanics (page 27); who having found that similar right-angled triangles represent the spaces described by a gravitating body in different times, infers that the spaces are as the squares of the times, &c.; whereas, conversely, his similar right-angled triangles represent the spaces described, because it is found by experiment and observation that the spaces described by gravitating bodies are as the squares of the times during which they gravitate; by which inverted method he seems to make the effect occasion its

cause.