...are the same eternal truths which God sees. For God sees as well as I that twice two are four, and that triangles on the same base and between the same parallels are equal. I can also discover, at least dimly, the relations of perfection among these ideas; and these relations...

...therefore the three parallelograms AUDC, BDFE, and EFHG are equal to each other. Cor. Hence it follows that triangles on the same base and between the same parallels are equal, since they are the half of the parallelograms on the same base and between the same parallels (by XXII)....

...solutions, and to point out the cases in which the full number cannot be obtained. PROPOSITION 37. THEOREM. Triangles on the same base and between the same parallels are equal to one another. Let ABC and DBC be As on the same base BC, and between the same ||s AD and BC, then...

...same ||8; and for similar reasons, a CDRS = a ABRS. .-. a ABPQ = a CDRS. Proposition 37. THEOREM — Triangles on the same base, and between the same parallels, are equal in area. Draw AX j| to BP, and BY || to AQ ; and let them meet PQ, produced both ways, in X and Y respectively....

...purpose is here given. This method is founded on a well-known proposition of Euclid, in which it is shewn that triangles on the same base, and between the same parallels, are equal. Let ABC, ABDbe triangles on the same base AB, and between the same parallels AB, CD; then the triangle...

...THREE HOURS. 1. Enunciate the propositions of Euclid dealing with the equality of triangles. 2. Prove that triangles on the same base and between the same parallels are equal. 3. If a straight line be divided into two parts, the square on the whole line is equal to the sum of...

...angle LMN to the angle PQR, also the side MN equal to QR; prove that the side LMis equal to PQ. 2. Triangles on the same base and between the same parallels are equal to one another. 3. In a right-angled triangle the square on the side opposite the right angle is equal...

...the parallelogram. 36. Parallelograms on equal bases and between the same parallels are equal. 37. Triangles on the same base and between the same parallels are equal. 38. Triangles on equal bases and between the same parallels are equal. 39. Equal triangles on the same...

...another. 36. Parallelograms on equal bases and between the same parallels are equal to one another. 37. Triangles on the same base and between the same parallels are equal to one another. 38. Triangles on equal bases and between the same parallels are equal to one another....

...quantity, treats almost exclusively of such abstract generalities; as 6=2x3; a?—y' = (x+y) (x—y); Triangles on the same base, and between the same parallels, are equal. § 128. Inference in the quantitative whole is immediate and mediate. Immediate inference from equivalent...