E. 1| Hyp. 2 Conc.

Let AE: EB = CF: FD.

then AE + EB = : BE = CF + FD: DF;

i. e., AB BE = CD: DF.

C. 1 Pst. 1, V. Of AB, BE, CD, DF take any equims. GII,

·.· KO, NP are equims of BE, DF, & KH, NM
also equims of BE. DF;

.. if KO > = or < KH, then NP > =
or > NM.

CASE 1. Let KO KH, & .. NP NM. Fig. 1 & 2.

D. 1 C. 1.

2 Ax. 3, V.

3 H. Ax. 3,V.

4 Sim.
5 H.

... GH, HK equims of AB, BE, AB being gr. than BE;

.. GH > HK,

but KO KH, .. GH > KO.

In the same manner LM > NP;

• KO KH, .. GH, a m of AB always> KO the same m of BE;

.. also LM the m of CD > NP the m of DF.

CASE II. Let KO > KH, and .. NP > NM. Fig. 3.

D. 1 C. 1.

25, V.

3 Sim.

4 5, V.

5] D. 2.

66, V.

7 C. 2.
8 Pst. 1, V.

9 6, 5.

SUBDIVISION 1

And the whole GH is the same m of the whole AB as HK of BE,

.. rem. GK same m of rem. AE as GH of AB. or as LM of CD.

LM the same m of CD, as MN of DF,

.. rem. LN same m of rem. CF as the whole LM
of the whole CD.

But LM same m of CD, as GK of AE;
.. GK same m of AE, as LN of CF,

i. e. GK and LN are equims. of AE & CF.
And ... KO, NP are equims. of BE, DF;
& from KO, NP may be taken KH, NM
also equims. of BE, DF,

.. rems. HO, MP either rems. BE, DF,
or are equims. of BE, DF.

Let HO, MP = BE, DF. Fig. 3.

D. 1 H& Case 2, D. 6. AE: EB = CF : FD, & GK, LN

SUBD. 2. Let HO, MP be equims. of EB, FD. Fig. 4.

D. 1 H. & Case 2,D. 6. AE: EB = CF: FD, and of AE, CF,

2 H.

3 Def. 5, V.

the equims are GK, LN,

& of EB. FD the equims are HO, MP;
.. if GK > = or HO, LN > =
or > MP;

4 Sub. & Ax. 5, I. but if GH > KO, from each taking KH,

then GK > HO;

51

6 Add. Ax. 4, I.

7

8 Sim.

.. also LN > MP;

to both add NM .. LM > NP;

.. if GH > KO, LM > NP.

So, if GH = or < KO, LM = or < NP;

9 Case I. D. 3, 4. And when KO KH, then GH > KO

10 C.

11 Def. 5, V.

12 Rec.

& LM > NP;

Otherwise. If A: BC: D ;- and of B, D any like pts, as the nth, be contained in A. C, m times exactly, or with like remainders; i. e. if m BA, & m D= = C, then those parts will be contained in A + B & C + D, the same number of times exactly, m+n with the same remainders;

.. by Def. 5,V. A+ BB = C+D: D.

COR, A A+B= C:C+ D.

:

USE & APPL. This method of reasoning is often employed. By an extension of it, as indicated in the Corollary, we find that the terms of a proportion are also proportional by addition. For, Let A:B= C: D,then, addendo, A: A + B = C: C + D;

Invertendo. B: A=D: C; Componendo, 18, V. A+B:A=C+D: C; .. invert. (A, V.) A : A + B = C: C+ D.

PROP. XIX.-THEOR.

If a whole magnitude be to a whole, as a magnitude taken from the first is to a magnitude taken from the other; the remainder shall be to the remainder as the whole to the whole.

"If a whole be to a whole as a part taken away to a part taken away; the part left is to the part left, as the whole is to the whole." EUCLID.

66

If four magnitudes, which are all of the same kind, be proportional, the first being greater than the third, and the second than the fourth; then the excess of the first above the third, shall be to that of the second above the fourth as the second is to the first."-HOSE.

DEM. 16, V. Alternando;-17; V. Dividendo; 11, V. Ratios the same to the same ratio are the same to one another.

E. 1,Hyp.

2 Conc.

Let the whole AB : the whole CD a pt. AE
from AB CE a pt. from CD; B.

then rem. EB: rem. FD = the
whole AB: the whole CD.

AB: CD = AE : CF,

.. alt. BA: AE=DC: CF;
Ms jointly are propl.,-
separately they are also;
DF: FC,

.. BE: EA =

15

6

D

10

E

F

9

D. 1 H. & 16, V...

2 17, V.

&

but A E: CF

=

6

6' Rec

teger AB integer CD.

.. If a whole magnitude, &c.

Q. E. D.

= AB: CD;

Otherwise,

A: B= C: D, C being < A & D < B ;

16, V. & 17,V. | .. alt. A : C=B : D, & div. A-C: C = B−D ; D.

D. 1

Hyp.

2

3

16, V.

COR. 1.

Again alt. A-C : B−D = C : D.

but A: B = C : D, .. A—C : B−D = A : B.

Also the remainder shall be to the remainder as the magnitude taken from the first to that taken from the other.

Or." the excess of the first above the third to that of the second above the fourth, as the third to the fourth.-HoSE.

A—C: B—D = A: B; and C: D = A: B;
.. A-C: B-D = C: D.

COR. 2. If any magnitudes, A, B, C, D &c. are in Geometrical Progression, i.e., A: B:C: D &c., the differences, A~B,: ~B:C CD &c,—will form a geom. progression, A~B: B~C : C~D &c.—the successive terms of which have the same ratio with the successive terms of the former.

A: B, & C : D = B: C &c.
A: B,

.. A~B: B~C =

& B~C: C~D = = B: C,

i.e., as A to B, i. e. as A~ B to B ~ C; and so on.

COR. 3. And conversely, any number of Magnitudes,, A, B, C, D &c. in geometrical progression, A: B: C: D&c,, may be considered as the differences of other magnitudes A, B′ C D' &c., forming a geometrical progression in which the first term A' is to A as A to A B, A' to B' as A to B &c., and the successive terms have the same ratio with the successive terms of the former.

2

For let a progression be taken in which A': A = A : A ~ B, and A': B' = A: B.