...triangle is half of a parallelogram on the same base and between the same parallels (41. 1.), and all triangles on the same base and between the same parallels are equal (37.1.); consequently ka b will denote the number of square feet in any plane triangle. Hence the surface...

...part of the base the perpendicular was taken ; since it has been shown (in cor. to theo. 1 3 geom.) that triangles on the same base and between the same...base the perpendicular was taken, in order to set it off in its due position ; and hence the map is easily constructed. PROBLEM XVI. PL. 8. fig. 5. To determine...

...THEOR. Parallelograms upon equal bases and between the same pa~ rallels, are equal. PROP. 37, THEOR. Triangles on the same base, and between the same parallels, are equal. Draw through one extremity of the base (to meet the line parallel, to the base,) lines respectively...

...occurred in the preceding books. 1. Thus it was proved in the 37th proposition of the first book, that all triangles on the same base, and between the same parallels, are equal in area : and hence, if only the base and area of a triangle be given, its vertex may be at any point...

...part of the base the perpendicular was taken ; since it has been shown (in cor. to theo. 13 geom.) that triangles on the same base and between the same parallels are equal i but if you would draw a map from the bases and perpendiculars, it is evident that you must know at...

...triangle constructed — scalenous triangle - - 34 Right angled triangle - - - - 36 Peculiar properties of triangles on the same base, and between the same parallels are equal - - 37 Similar triangles, what — their sides proportional - 37 The longest side opposite the greatest...

...in the preceding books. 1. Thus, it was proved in the 37th proposition of the first book, that all triangles on the same base, and between the same parallels, are equal in area: and hence, if only the base and area of a triangle be given, its vertex may be at any point...

...therefore the three parallelograms ABDC, 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 ol the parallelograms on the same base and between the same parallels (by Art....

...BOB. The former of the above propositions seems to be a proper generalization of Euclid's theorem " that triangles on the same base and between the same parallels are equal." If we have any number of surfaces of definite extent, either plane or curved, and we form pyramids...

...angles of a triangle are together less than two right angles. (See EUCLID, Book I. Prop, xxii.) 6. Triangles on the same base and between the same parallels are equal. (See EUCLID, Book I. Prop, xxxvii.) 7. If a parallelogram and a triangle be on the same base, the parallelogram...