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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.
“ Cor. If we take A and B, as above, and m B> A, there results B> ; much more, then, shall the inequality B am 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, f is indefinitely greater than '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,
B C, 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 DE=AD, 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 BC.
“ For the opposite vertical angles A D B, CDE are equal (Simson's Euclid, B. I., prop. 15), and by construction B D=DC, AD =DE; hence (Sim., B. I., p. 4) the triangles A DB, CDE are identical ; so that their bases A B, C E are equal, and their angles A B D, EC D are equal; also the angle BAD equals the angle CED. Hence the angle C of the triangle ACE 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 (A BC); .. the sum of the angles of the triangle ACE is equal to that of the given triangle A B C, 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 CA E is not greater than CEA (Sim., props. 5, 19, B. I.); hence the angle A of the triangle ACE is not greater than f; we shall call the triangle A CE the first derived triangle. Of the two sides A C and CE of the triangle A CE, let C E be that which is not the greater ; and let a right line be drawn from the angle A, opposite to the side C E, 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 A B C, 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 A B C. And if A C is not greater than C I, then it may be shown (as before) that the angle I is not greater than the angle CAE-2, and since C A E is not greater than Î, .. the angle I is not greater than 2, we shall call A C I the second derived triangle.
“We may in the same way (that we derived the triangle A C I from AC E) derive a triangle from ACI (called the third derived trian. gle), having the sum of its angles equal to that of ACI, 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