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XXXIII.

GEOMETRICAL

THEOREMS.

I. If a Right Line do fall upon Two Parallel Right Lines, it
maketh the Alternate Angles Equal.

Fig... Lines and NO,
O the Right Line PQ, falling upon the Two Parallel Right
Lines L M and N O, doth make the Alternate Angles Equal.
As the Angle PR M, equal to the Angle NSQ; and PRL,
equal to QSO, Elem. L. 1. P. 29.

II. If divers Right Lines, be cut by divers other Right Lines, which
are Parallel one to the other, the Segments are Proportional.

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.

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IV. If Two Rectangled Figures, be made of One of the Sides of that
Figure, and of any Two Segments of the other Side of the fame
Figure; thofe Two Parallelograms added together, shall be equal
to that Figure.

So if the Parallelogram GLIM be made of the Side B 9, and
a Segment of the other Side; namely, 4, that Parallelogram fhall
produce 36, for 9 times 4 is 36. And if another Parallelogram
ihall be made of the whole Side B 9, and the other Segment of
the other Side A7, namely, 3, they will produce the other Paral-
lelogram LHMK 27, for 3 times 9 is 27: And these Two
Parallelograms 36 and 27 added together, do make 63 equal to
the first.

V. If Four Right Lines be proportional, (that is, as the firft is to
the fecond, fo is the third to the fourth) the Right-angled
Figure made of the Two Means, (or Middle Terms) hall
be equal to the Rectangled Figure, made of the Two Extream

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
them be given, the Fourth is also given. For,

The Rectangle Figure made of the Two Means, being divided
by one of the Extreams, the Quotient fhall be the other Extream.

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.

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Fig.

CONSECTART II.

Hence alfo it followeth, that equal Rectangled Figures, have their
Sides reciprocally proportional: That is,

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,

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So is

YE to RS 4 :: So is 12

8:

RS: to XZ

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VI. If Three Lines or Numbers be proportional, (that is, as the
Firft is to the Second, fo fhall the Second be to a Fourth) the
Square made of the Means, is Equal to the Oblong made of
the Extreams.

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

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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

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