Sidebilder
PDF
ePub
[ocr errors][merged small][merged small][merged small][merged small][merged small][merged small]
[ocr errors]
[ocr errors]

PROPOSITION VIII. THEOREM VII. F a point (D) be taken without a circle (BGCA), & straight lines (DA, DĖ, DF, DC,) be drawn from it to the circumference, whereof one (DA) passes thro' the center (M); of those which fall upon the concave circumference, the greatest is that (DA) which passes thro' the center; & of the rest, that (DE or DF) which is nearer to that (DA) thro' the center, is always greater than (DF or DC) the more remote: but of those (DH, DK, DL, BG,) which fall upon the convex circumference, the feast is that (DH) which produced passes thro’ the center : & of the rest, that (DK or DL) which is nearer to the least (DH) is always less than (DL or DG) the more remote: & only two equal straight lines (DK, DB,) can be drawn from the point (D) unto the circumference, one upon each side of (DH) the least. Hypothesis.

Thesis.
1. The point D is taken without a 1. DA which passes thro' the center M is
O BGCA in the same plane. the greatest of all the straight lines

DA, DE, DF, DC.
II. The Araight lines DA, DE, II. DE or DF, which is nearer to DA is >

DF, DC, are drawn from this DF or DC, the more remote.
point to the concave part of III. DH which when produced passes thro
tbe O BGCA.

center M is the least of all the straigbe

lines DH, DK, DL, DG.
III. And those straight lines cut the IV. DK or DL which is nearer to the line

convex part in the points H, DH, is < DL or DG the more remote.
K, L, G.
V. From the point D only two equal

fraight
lines DK, DB, can be drawn, one uponi

each side of DH the least.

1. Preparation.
Draw the rays ME, MF, MC, MK, ML.

DEMONSTRATION.
H E two sides DM+ ME of the ADME are > the third DE, P. 20. B. 1.
And because ME = MA (D. 15. B. 1.).

[ocr errors]

2. DM + MA or DA will be > DE.

It is demonstrated afer the same manner that:
3. The straight line DA, which passes thro’ the center M, is > any

other straight line drawn from the point D to the concave part of
the O BGCA.

Which was to be demonstrated I.
Moreover, DM being common to the two ADME, DMF, ME

MF (D. 15. B. 1.), & V DME > V DMF (Ax. 8. B. 1.). 4. The base DE will be also > the base DF.

P. 24. B. 1. In like manner it may be shewn that: 5. The straight line DF is > DC, & fo of all the others. 6. Consequently, the straight lines DE or DF, which is nearer the line DA, which passes thro' the center, is > DF or DC more remote.

Which was to be demonstrated. II. 7. Again, the sides DK + KM of the ADKM are > the third DM. P. 20. B. 1.

If the equal parts MK, MH, (D. 15. B. 1.) be taken away. 8. The remainder DK will be > DH, or DH < DK. It may be proved in the same manner,

that: 9. The straight line DH is < DL, & so of all the others. 10. Consequently, the straight line DH, which produced passes thro' the

center M, is the least of all the straight lines drawn from the point D to the convex part of the O BGCA.

Which was to be demonstrated. III. Also, DK, MK, being drawn from the extremities D & M of the side

DM of the A DLM to a point K, taken within this A (Hyp. 3.). 1. It follows, that DK + MK< DL + ML.

P. 21. B. L. And taking away the equal parts MK, ML, (D. 15. B. 1.). 12. The straight line DK will be <DL.

In like manner it may be shewn, that : 13. The straight line DL is < DG, & fo of all the others. 14. Consequently, the straight lines DK or DL, which are nearer the

line DH, which produced passes thro' the center, are < DL or DG the more remote.

Which was to be demonstrated. IV.

II. Preparation 1. Make V DMB = VDMK, & produce MB 'till it meets the O. P. 23. B. 1.

2. From the point D to the point B, draw the straight line DB. Pof. 1. Then, the side DM being common to the two ADKM, DBM, the side

MK = the fide MB (D. 15.B.1.), & VDMK= DMB (II. Prep.1.). 15. The base DK will be = to the base DB.

4. B. 1. But because any other straight line different from DB, is either near

er the line DH or more remote from it, than DB. 16. Such a straight line will be also < or > BD ( Arg. 14.). 17. Wherefore, from the point D, only two = Itraight lines DK, DB, can be drawn, one upon each side of DH.

Which was to be deinonstrated. V.

P.

[blocks in formation]

PROPOSITION IX.

THEOREM VIII.

If a point (D) be taken within a circle (ABC)

, from which there fall

more than two equal straight lines (DA, DB, DC,) to the circumference ; that point is the center of the circle. Hypothefis.

Thesis, From the point D, taken within a O ABC, The point D is the center of there fall more than two equal fraight lines the O ABC. DA, DB, DC, to the O ABC.

DEMONSTRATION. If not,

Some other point will be the center.

Becau

ECAUSE the point D is not the center (Sup.), & from this point D there tall more than two equal ttraight lines DA, DB, DC, to ihe O ABC (Hyp.). 1. It follows, that from a point D, which is not the center, there can

be drawn more than two equal straight lines ; which is impossible. P. 7. B. 3. 2. Consequently, the point D is the center of the O ABC.

Which was to be demonstrated,

[blocks in formation]
[ocr errors]

PROPOSITION X. THEOREM IX. . NE circumference of a circle (ABCEG) cannot cut another (ABFCG) in more than two points (A & B). Hypothesis.

Thesis. The two O ABCEG, ABFCG, cut

They cut one another only in two one a nother.

points A & B. DEMONSTRATION.

If not,

P. 1. B. 3.

Pof. 1.

They cut each other in more than two points, as A, B, C, &c.

Preparation.
1. Find the center D of the O ABCEG.
2. From the center D to the points of section A, B, C, &c.

draw the rays DA, DB, DC. Because

ECAUSE the point D is taken within the O ABFCG, & that more than two straight lines DA, DB, DC, drawn from this point to the circumference of the O ABFCG, are equal to one another, (Prep. 1. & D. 15. B. 1.). 1. The point D is the center of this O.

But this point D being also the center of the O ABCEG (Prep. 1.). 2. It would follow, that two O ABFCG, ABCEG, which cut one ano

ther, have a common center D ; which is impossible. 3. Consequently, two O ABCEG, ABFCG, cannot cut one another in more than two points.

Which was to be demonstrated.

P. 9. B. 3.

P. 5. B. 3

[ocr errors]
[merged small][merged small][ocr errors][merged small][merged small][merged small]

I

PROPOSITION XI. THEOREM X.
F two circles touch each other internally in (A); the straight line which
joins their centers being produced, shall pass thro' the point of corta& (A).
Hypothesis.

Thesis,
The ftraight line CA joins the centers of This straight line CA being pro-
the two O AGE, ABF, which touch

duced, polles tbro' the point of each other internally in A.

contadt A of those two O.

DEMONSTRATION.

If not,

The straight line which joins the centers, will fall otherwise, as
the straight line CGB.

Preparation.
From the centers C & D to the point of contact A, draw the
lines CA, DA.

Pos. 1.

BECAUSE in the A CDA, the two sides CD & DA taken toge

A ,
ther, are the third CA (P. 20. B. 1.), & that CA: CB (D. 15. B. 1.).
1. The straight lines CD + DA will be allo > CB.

Therefore, if the common part CD be taken away from both sides.
2. The straight line DA will be > DB.

Ax. 5. B.L.
But the itraight line DA being = to DG (Prep. & D. 15. B. 1.).
DG will be also > DB, which is impossible.

Ax. 8.B... 4. Wherefore, the straight line CA, which joins the centers of the O

AGE, ABF, which touch each other internally, being produced, will
pass thro' the point of contact A.

Which was to be demonstrated.

1

« ForrigeFortsett »