Fig.

XXXIII.

Fig.

GEOMETRICAL

THEOREMS.

S

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

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.

Let the Two Lines RS and RT, be cut by the Four Parallel XXXIV. Lines V, X, Y, Z. I say then, the inter Segment Ra and Rb; as alfo aT and b S, are proportional one to the other: For if R a be one third Part of RT, Rb shall be one Third of R S, &c. because the Right Line a X, cutteth off one third Part of the Space ÆV TZ, and therefore it cutteth off a third Part from every Line drawn within that Space.

Fig.

III. If Two Right Lines be multiplied into one another, there is
made of them a Right Angled Parallelogram.

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 CDEF.

00

IV. If

!

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; those Two Parallelograms added together, shall be equal
to that Figure.arm

So if the Parallelogram GLIM be made of the Side B 9, and a Segment of the other Side; namely, 4, that Parallelogram shall produce 36, for 9 times 4 is 36. And if another Parallelogram shall be made of the whole Side B9, and the other Segment of the other Side A7, namely, 3, they will produce the other Parallelogram 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 first is to the fecond, fo is the third to the fourth) the Right-angled Figure made of the Two Means, (or Middle Terms) Shall be equal to the Rectangled Figure, made of the Two Extream

Terms.

Let there be Four Proportionals; as N 4, 06, P8, Q12. I Fig. say, that the Right angled Figure RST V, made of the Two XXXVI. Means, O 6, and P 8, shall be equal to the Rectangled Figure XYZÆ, made of the Two Extreams, N 4, and Q 12: For, as 6 times 8 is 48, so 4 times 12 is 48 also. And from hence will follow

8. CONSECTARY 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 shall be the other Extream. -As the Rectangle Figure, made of the Two Means 6and 8, is 48, this divided by 4, the Leffer Extream, shall give you in the Quotient 12, for the Greater Extream. Or, if the proportional Terms given had been (by Transposition) 4, 8, 6, 12, the Rectangle 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.

CONSECTARY ΙΙ.

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; so is the Greater Side of the Second, to the Greater Side of the First, & Contra. As in the Equi-rectangular Figures RSTV, and XYZE, appeareth: For,

So is

VI. If Three Lines or Numbers be proportional, (that is, as the
First is to the Second, So shall the Second be to a Fourth) the
Square made of the Means, is Equel 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 also the Oblong made of the Two Extreams 4 in 16 is 64 also.

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 Base TS, it cutteth off from the Side RS one third Part, and it cutteth from the Side RT one third Part also: And so they shall be proportional by the second hereof. For,

As RT: to RS ::
As Rc: to cT::
As Rc: to Rd ::

So is Rc: to Rd. And
So is Rd: to d S. Alfo,
So is c T : to d S.

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,

1

Fig. (1.) Let ABC and ADE be Two Plain and Equiangled XXXVII. Triangles: So that the Angles at B and D, at A and A, and

4