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# Title: Lexicon Technicum: Or, An Universal English Dictionary Of Arts And Sciences (shelfmarks Leibn. Marg. 71; ZEN Leibn. Marg. 71)
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Since cheretore OurhrredS Theoremm, 25 is Tev-
Fior 12, Their Summ, caken trom 90, che summ
cd abore, is, Fhat SS = DD = AR. Where oEcheir Squares, lcares 7B; and added 10 27, cheir
Core, : divide all b7 +: 55 - DD = R 5 Xet
YR De made a Fangent ro the leNer, Or 2 RiglIE.
sine ro the srearer Circle, as rou sec in the Figure
accordins a5 D, Ox S, is siven: And deaw alsorhe
Hrporhenuse II. Then Will che Base ot the Tri-
onsle be d. And HHI - d d = R (b7 47. C L.)
That is,
DD
P. Wheretore BY Fran-
Cposirion, IL FF = R. j- d d, and cheretore HI =
: R -= d d. And consequentir, il ic had been
in the sirst or Second Forms, Wvhere d and R evere
siven, EI vill also be Found. Or it II had been
diren, and d required asinche Third Foem. Since
AHI = R - ] dd: Theretore, LIHI -. R = d d,
And V: HIHI = R = d: And harins thus found A
and a, the = Sum and = Ditterence oF the trro ROOrS.
Then HI = a (-= 0 p) Wvill bc che Greatex ROOE a.
and II a (= p n) Wvill be che lesser, wtich vil
be Afirmarire or Negarire, according eo the Form
and Circumstances oEche Equarion.
A Ouestion and Problems in Adfected Quadratick
Equations.
QVESTTON.
Euo Mlen have each a certain Number os Cronns
ohose sum Subtracted from ehe Sum of their
Squares, leaves R =78: But their Sum added +0
males 39 = 5.
ITon mans Cromnshad each:
For the untenoun Sum os che Numbers put 2 a.
And For their Disserence 2 c.
For then ehe Numbers may be thus Nored
 e = = and a = c
Where = and =siexisie che prearer aned lester
C
Then,
2 a a =  2 C e = Sum Ot CCH AI.
2 4 4 -] - 2 cc -= 25 == R. byche
Srare oErhe Ouection.
2 -- 2 3 a a - ec - a  -. 39. =
S.
by TratP. 41 59 ad  a = cc. WMCH SuP
Dy Tranc. 1
e
Wwill ac last help ro And c.
a a = ec - = 2 a = 5.
Their
eroduc added ro their Sum.
a a - 2 a - 5 =
ee
39 = a a -) a = a
OY TratIP. 8(18 = 2 a 4 -  a
3) a 4 -  = a = 3 = S, Whic
Ouadrarick oF hC HIILLOLLIL
Compl. 10 a a - = a 17 25 = 39- 25
221 et 1. 37 1 260
Rectans, makes 39.
Let ci cooLtuldtosoltieus lillcead et a and
The Difference of both the Less of a
Rishransled Friansle beine siver
from the Hypothenuse; 10 find the
Sides and the Friancte severally,
and to Form it.
Tetthe Difterence ot thelellet Side Fromthe LIy.
Pochenuse be (b) and char oE the grearer (d.)
For ce Prearer, Side Conehr pur (a.)
Thet vilt,
a-) d = Fyporhenuse, and
a- d= b= bc thelesser Side
a a -) 2 a d - d d.=
2 a a
22d 2 a b- 2 bd.) d d 1 bb.
a 2 - 2 a b = 2 b d -   b0 = 0
by Comparison and Transposi-
tion oEche laA Scep
a a = 2 ab == 2 b d =- b b. votrick
is a Ouadraricke Equarion oErhie
Second Form.
tdus de te blcalet dide toul. d Co be C
2
7
LV. B. By this Method oF purtins a l C aliet d = e
c as I he Common was; many Ouestions
producins Adtected Ouadrarick liquarions.
Fspecially vrtien the sum and Diferenc, or
Sum or Disterence ot che ascates et lle Man-
rities Touslie, are amons ThC Data.
otien etiat ar manae d, mas be solred as ca
SilY, and in the mannet ot simple Equations.
e.
- 2a
D+ 2a
L ICOBL E M I.
a a = 2 a b - - b b = 2 b d.
a =- b = v : 2 b d
a = V. 20u- b.
Geometrical Construction.
Trans
Comp
Find a mean Proportioras Betleen d a, Woset
Iet DC DF: xo Which, place at Righranoles FC
EOD F, and cuF Of II D = GD. Then wil
produced eo C (Corhar C H = F. D) will give CD
AC) che lesser Side oF che Priansle required
FOF a - ) d- b = L. Draw a Semicirele on C B
And (a) beins knOn, the ralue OF (e) ill and axpir A B FIB. Then drav A C and rhé
Lhanele is tound, wrhichis AC B.
2 a =- ~. 39
13lLhetetore 2 a = 22.
Number aS a  e, Thatis 9; and the lesler Was
=e; charis 3: Which Numbers 3 and 9, vili
Antiver the Ouestion.
bo found trom thefourth Scep. Where e = 2
NTou, by our Supposition al Aest, the GrCateF