Two hundred and ninety-fourth Meeting.

April 6, 1847.-MONTHLY MEETING.

The PRESIDENT in the chair.

Professor Strong, of New Brunswick, New Jersey, communicated the following papers, viz. :—

I. "An attempt to prove that the sum of the three angles of any rectilineal triangle is equal to two right angles.

"Def. Two quantities are said to be of the same kind, when the less can be multiplied by some positive integer, so as to exceed the greater. Thus, if A and B are quantities of the same kind, and if A is greater than B, then some positive integer, m, may be found, such that the inequality m B➤ A shall exist. For if m is taken greater than the quotient arising from the division of A by B, then evidently there results the inequality m B>A, as required.

"Dem. If m denotes any positive integer, then shall the inequality 2">m obtain. For the first member of the inequality denotes the product arising from taking 2 as a factor as often as these units in m, whereas the second member is the sum of the units represented by m, and the inequality is evident.

A

"Cor. If we take A and B, as above, and m B>A, there results B>; much more, then, shall the inequality B> have place. This follows at once since it has been shown that 2" is greater than m; and it is evident that if m is an indefinitely great number, is indefinitely greater than

A

m

"Ax. No angle of a rectilineal triangle can exceed two right angles. "Prop. 1. To find a triangle that shall have the sum of its angles equal to the sum of the angles of any given triangle.

"Let A B C denote the given triangle; and suppose one of its sides,

BC, is bisected at D, and that D and the opposite angle A are connected by the right line A D, which is produced in the direction A D

to E, so that DEAD, and that the point E and the angle Care connected by a right line; then shall the sum of the angles of the triangle AC E equal that of the triangle A B C.

"For the opposite vertical angles AD B, C DE are equal (Simson's Euclid, B. I., prop. 15), and by construction BDD C, AD =DE; hence (Sim., B. I., p. 4) the triangles AD B, C D E are identical; so that their bases A B, CE are equal, and their angles ABD, ECD are equal; also the angle BAD equals the angle CED. Hence the angle C of the triangle A CE equals the sum of the two angles B and C of the given triangle (A B C), and the sum of the angles A and E of the triangle ACE is equal to the angle A of the given triangle (ABC); .. the sum of the angles of the triangle ACE is equal to that of the given triangle ABC, as required.

"Cor. 1. Let the angle B A C of the given triangle be denoted by A; then if CE (= A B) is not greater than A C, the angle CAE is not greater than CEA (Sim., props. 5, 19, B. I.); hence the angle A of the triangle A CE is not greater than ; we shall call the triangle ACE the first derived triangle. Of the two sides AC and CE of the triangle A CE, let CE be that which is not the greater; and let a right line be drawn from the angle A, opposite to the side CE, through the point, H, of bisection of C E, and suppose the line thus drawn to be produced in the direction A H to I, so that HI= HA; then connect the point I and the angle C of the triangle A CE by a right line; and there will be formed the triangle A CI. In the same way that it was shown that the sum of the angles of the triangle ACE is equal to the sum of the angles of the (given) triangle ABC, it may be shown that the sum of the angles of the triangle A CI is equal to that of ACE; consequently the sum of the angles of ACI equals that of the given triangle ABC. And if AC is not greater than CI, then it may be shown (as before) that the angle I is not greater than the angle CA E÷2, and since CA E is not greater than 4,.. the angle I is not greater than we shall call A CI the second derived triangle.

A

А 229

"We may in the same way (that we derived the triangle A CI from A CE) derive a triangle from ACI (called the third derived triangle), having the sum of its angles equal to that of A CI, and of course equal to that of the given triangle A B C, and having one of its angles not greater than the angle AIC÷2, and consequently not greater