PROP. XCIV.

See Scheme. 92.

If a Space AC be contained under a rationall lineTM AB and a third refiduall AD (AE-DE) the right line TS containing in power the space AC is a fecond medial! refiduall line.

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As in the former,TO and SO are . Therefore hyp, because DE a is AB. b the rectangle DI, and 22.10. fo DK,or TOS,fhall be uv.d therefore TSAC 76.10. is a fecond mediall refiduall.w.w.to be Dem.

PRO P. XCV.

See Scheme 92.

If a space AC be conteined under a rationall line AB and a fourth refiduall AD (AE-DE) the right line TS conteining the Space AC in power, is a Minor line.

C

C

As before, TOSO. Therefore because AE a lem.91.10. b byp. bis AB, fhall AI (TOq+SOq) bepr.but, as 20.10. before,the rectangle TOS is v. therefore TS/d77.10. AC is a Minor line. w.w.to be Dem.

If a fpace AC be contained under a rationall line AB and a fift refiduall AD (AE DE) the right line TS containing in power the space AC, is a line which maketh with a rationall space the whole space mediall.

a byp.

For again TO SO. therefore fince AE a is ABb alfo Al, that is TOq+ SOq fhall be b 22.10. v.But,as in the 93.the rectangle TOSis pve whence TS AC is a line which with pv makes a whole ur. W.W.to be Dem. AMTOZO

"

.C

ICA PROP.

PROP. XCVII.

FGE

if a Space AC be contained under a rationall line AB, and a fixt refiduall AD (AEDE) the right line TS containing in power the space AC, is a line making with a

BL CNKHI mediall rectangle, a whole

T

alem-gt.

20.

679, 10.

Space mediall

As often above, TOT SO. alfo, as in 96, TOq+ SOq is r. but the rectangle TOS is pv, as in 94. a Laftly TOq+SOqTOS. therefore TS=✔✅ AG is RM a line which with makes

a whole v. W.W.to be Dem.

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47.ax. 1. 81.6.

E

DK is

Vpon a right line DE LDF = ABq, and DH apply the rectangles

И HK

ACq, and IK = BCq. and let GL be bifeated in M, and the line MN drawn parallel to GF.

Then 1. the rectangle ACq+BCq. as the conftru&tion manifefts. 2. The rectangle ACB GN or MK. For DK ACq+BCqb2 ACB + ABq. but ABq "DF. therefore GK c 2 ACB. and confe quently GN or MKACB,

3. The rectangle DIL MLq. For because ACq. ACB; ACB. BCq, that is DH, MK:: MK,

IK. ethence is DI, ML:: ML. IL. ftherefore DIL f 17 6. =MLq.

4. If AC be taken BC, then DK fhall be ACq. For ACq+BCq (DK) & Acq.

1

g 16, 10,

5. Likewife DLDLq GLq. For becaufe DH (ACq) TL IK (BC) b thence fhall DI 10.10. IL. therefore✔ DLq — GLq T

be DL.

6. Also DL

ACB. that is, DK
GL.

k18. 10.

1 lem.26.10.

GL. For ACq+ BCq/2 m 10. 10.
GK. therefore DL

7. But if AC be taken ✓DLq-GLq.

A

D

BC, then DL fhall be

PROP. XCVIII

.

The fquare of a refi-
C duall line AB (AC -
LBC) applied to rationall

line DE, makes the
breadth DG a firft refi-
duall line.

Doe as is enjoined
in the Lemma next pre-

E F NHK ceding. Then because AC,BC,4 are. balfo DK (ACq++BCq) fhall be ACq. therefore DK is pv. d wherefore DL is pDE. e Likewise the rectangle GK (2 ACB) is v. ftherefore GL is p' DE.g and confequently DLL GL. But DLq GLq. therefore DG is a refiduall, and that of the first order (becaufem AC BC, and therefore DL DLq- GLq. w.w.to be Dem.

23.10.

83. 10. hsch.12. 10. * 74.10. 1.def.85.

10.

miem.97.

10.

PROP.

PROP. XCIX.

See the following Scheme.

The fquare of a firft mediall refiduall line AB (AC
-BC) applied to a rationall line DE,makes the breadth
DG a fecond refiduall line.

Suppofing the foregoing Lemma; because AC and
BC are, thence fhall DK (ACq +BCq) be
TLACq. c wherefore DK is ur. d therefore DL is p
DE. e alfo GK (2 ACB) is pv. ftherefore GL
is DE;g wherefore DLL GL. b But DLq
GLq, therefore DG is a refiduall line ;
caufe DL is DLq GLq, therefore fhall
DG be a fecond refiduall. W. W.to e Dem.

k

and be

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10.10.

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DE. alfo DGK is av

applied to a rationall line DE makes the breadth DG a third refiduall line.

Again DK is μv. a wherefore DL is a whence GL is e

DE. b likewife DK GK. ewherefore DĽ

dfeb.12. 10. GL. but DLq GLq. e therefore DG is a refi3.def.85, duall line, and that off the third order, g because DL DLqGLq. w.w.to be Dem.

10.

glim.97.10.

PROP. CI.

See the foregoing Scheme.

The fquare of a Minor line AB (AC-BC) applied

to

toa rational line DE, makes the breadth DG a fourth refidual.

*byp

As before,ACq+BCq, that is DK,is pr. there-1.10, fore DL is TLDE. but the rectangle ACB, and b123 10. fo GK (2 ACB) * is ur. b wherefore GL is 13.10, DE. therefore DL GL. but DLq GLq. elem.97. dfeb.1. and because *ACqL BCq, ethence fhall DL be 10. f 4 def.88. ✓DLq - GLq. ftherefore DG ha's the con- 10. ditions required to a fourth residuall.w.W.to be Dem.

C

The fquare of a line AB (AC BC) which makes with a rational space the whole space mediall, ap plied to a rationall line DE, makes the breadth DG a fift refiduall line.

For, as above, DK is ur. a wherefore DL is DE.alfo GK is pub whence GL is TLDE there

fore DLL GL. d but DLqL GLg. Moreover DLL✓DLqGLq. wherefore DGf is a fift refiduall. w.w.to be Dem.

The fquare of a line AB (ACBC) making with a mediall space the whole space mediall, applied to a rationall line DE,makes the breadth DG a fixt refiduall line.

As above DK and GK are pa;
DE,also DK 6

and GL are ¿

fs.def.85.

wherefore DL
GK.c whence by and
Tem.97.10%

DL GL. therefore DG is a refiduall. b And c 10.10.

874.10.

e

whereas ACqBCq. and fo DL √ DL9 078 85. GLq, e therefore DG fhall be a fixt refiduall. 10. w.w.to be Dem.

3