XXXIII. GEOMETRICAL THEOREMS. I. If a Right Line do fall upon Two Parallel Right Lines, it Fig... Lines and NO, II. If divers Right Lines, be cut by divers other Right Lines, which a Fig. Let the Two Lines RS and RT, be cut by the Four Parallel XXXIV. Lines V, X, Y, Z. I fay then, the inter Segment R and R b; as alfo aT and b S, are proportional one to the other: For if R a be one third Part of RT, Rb fhall be one Third of R S, &c. becaufe the Right Line a X, cutteth off one third Part of the Space AVTZ, and therefore it cutteth off a third Part from every Line drawn within that Space. III. If Two Right Lines be multiplied into one another, there is made of them a Right Angled Parallelogram. Fig. Let the Two Sides to be multiplied be A 7 and B 9. If of the XXXV. Lines. A and B, a rectangled Parallelogram be made, it will be. fuch a Figure as CDEF: Or, if A 7 and B9, be multiplied. each by other, the Product will be 63, and fo many little Parallelograms are there contained in the larger Parallelogram C D E F. IV. If Two Rectangled Figures, be made of One of the Sides of that So if the Parallelogram GLIM be made of the Side B 9, and V. If Four Right Lines be proportional, (that is, as the firft is to Terms. Let there be Four Proportionals; as N4, 06, P 8, Q12. I Fig. fay, that the Right angled Figure R S T V, made of the Two XXXVI. Means, O 6, and P 8, fhall be equal to the Rectangled Figure XYZE, made of the Two Extreams, N 4, and Q12: For, as 6 times 8 is 48, fo 4 times 12 is 48 alfo. And from hence will follow CONSECTART. I. If Four Right Lines (or Numbers) be proportional, if Three of The Rectangle Figure made of the Two Means, being divided As the Rectangle Figure, made of the Two Means 6 and 8, is 48, this divided by 4, the Leffer Extream, fhall give you in the Quotient 12, for the Greater Extream.Or, if the proportional Terms given had been (by Tranfpofition) 4, 8, 6, 12, the Rectan gle made of the Means 8 and 6 would be 48; which divided by 12, the Greater Extream, would give in the Quotient 4, the Leffer Extream. Fig. CONSECTART II. Hence alfo it followeth, that equal Rectangled Figures, have their As the Leffer Side of the First Figure, is to the Leffer Side of XXXVI, the Second Figure; fo is the Greater Side of the Second, to the Greater Side of the First, & Contra. As in the Equi-rectangular Figures RST V, and XYZE, appeareth: For, So is YE to RS 4 :: So is 12 8: RS: to XZ VI. If Three Lines or Numbers be proportional, (that is, as the As let the proportional Numbers be 4: 8: 8, the Proportion will be As 4 is to 8: So is 8 to 16. So the Square of the Two Means 8 is 64: So alfo the Oblong made of the Two Extreams 4 in 16 is 64 alfo. VII. In a Plain Triangle, a Line drawn Parallel to the Base, cutteth the Sides thereof proportionally. Fig. As in the Plain Triangle RTS (in the Scheme of the Second XXXIV. hereof) if c d be Parallel to the Bafe TS, it cutteth off from the Side R S one third Part, and it cutteth from the Side R T one third Part alfo: And fo they shall be proportional by the fecond hereof. For, As RT to RS: So is Rc to Rd. And VIII. If divers Plain Triangles be compared together, All Equiangled Triangles have the Sides about (or containing) the Equal Angles proportional: And the contrary, Eucl. Lib. 6. P. 4. For Illustration, Fig. (1.) Let A B C and A D E be Two Plain and Equiangled XXXVII. Triangles: So that the Angles at B and D, at A and A, and alfo |