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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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Series of che Progrestion is carried on. Thus che) And ao I the FOrmex RanK OL NNUMberS 2 WV 
Sum ot Phree LCttS, 1, 4, 9, 14, Llo  pne ncon Mcspitet, in chis 2 is che Common
27, 25 Y 5 OF F 28, OF Y -)
- 1= = 1. 10 3,
(diridins boch sides by 9) che Sum oF Six Lerms,
An 4, 9, 16, 25, 36, 9c. v1Z. 9F 10 SIX MMC
36, 1. c. 10 216, (diridins boch lides bY 72) 15
15 1 - . 2. -1- - = 103; and che Sum of Tvelre
Terms 650, 10 12 times 144, 1. c. 1728, (diri
dins boch lides by 576) 15 25 1 - - = --
03,
C5c. the Fraction adherins co them thus constaney
decreasins; fome by cheir TTaIE parES, Orhers by
Threc quarrers: (FOr = I5 = Z; therefore che ArI
Dirilor; Or, Wrhichis in effect che lame, the Com-
ion MuluiplicE 15 =.
ThC MIOIE Nacural and Simple Geomerrica) Pro-
srestion, is when ic begins wich Cnits; 25,
Decrement i5 = 3, and = = I5 ,3: cheretore che se- Former Progression, ic is in effect bur such a Pro-
Cond Decrement is = 2 (90.) Wherctore che sum
ot che lmce Prosrection will be coche Sum ol
che like Number ot Lerms equal tothe sredtet;
5,
grection as thislarter, mulciplied inco chac ArA Perm
2II alon8.
1. 1. 1. M. M. I. aC.
When ie begins with any orher LCHM, as a inche
LADEOSITION I
IE chere are Threc Ouanricies continually PrO
1s. 66. 25 2 Bc. 10 3, bY C0.
roil. 2d and 4eh, in Proeression Ceometrical.
Buc ihe Sum oF a Friplicare Arichmetical PrO.
prestion, (i. c. ascendins by che Cubes ol che C ae
dinal Numbers) proccedins From F ThrO 27, 64,
E8C. ad infinitum, is subquadruple oE the like Num-
ber oF Terms equal co the greatest: For che Sum
OE FOur TerMS, 1, 8, 27, 64, 1. c. 100, 10 4 times
64, 1. c. 256, (dividins boch sides by 64) Wvill
be tound 1o be a5 r ) = -]- - 5 10 4; bur ehe Sum
OF 8 TermS, 1, 8, 27, 64, 125, 216, 343, 512,
1. c. +296 10 8 rimes 5T2, 1. c. 4096, (dividins
boch sides bY 1024) Will be tound 10 be 25
i -f = 10 4, (9c. The adherins FractionS
thus constaney decrealins, che one by cheir HTaIE
Parr, che ochers by Three quarrers, (For 15 15 4,
and= 15 -7, 5c. Wheretore the Sum of cheln.
finite Prosrettion will be coche Sum oF a like (In-
tinite) Number oE Terms, equalto che grearest;
(90. = 335, (90. 10 4, b7
Coroll. 2d and A1t, 25 7 20 4. 8. F.. D.
Portional, the Reclansle ot the Eixtremes is equar
roche Square oF the Mean Frem.
TEs plain char the Fxrremes a, rra, mulciplied inco
one anocher, are equal co rhe Mean ra mulciplied
into it selF; Ox in Numbers, chat 3 times r2 is
equat t0 6 rimes 6.
the lame Conrinual Proporion a5
EROORESSTONI Beomerrical, or Beometricat Ihe Rectanoles oF che Extremes made erols- ways
Proportion Continued, is When Dumbers, Ox orher
Duanciries, proceed by equal Proportion Or Ratio s,
(property Cocalled) chat is, accordins ro onc Com-
mon Mustiplier, Ox Exponent ot che Common Ra
tio Whether Increatins Or Decreasine. AS,
Hc equal 10 che Rectanse ot he Meaa Lerme
beins erery way rrab, OF 48.
Surface of a Rsht Cone i5 equat 10 a CHcte obost
Radius is a Mean Proporcion Derveci ie stat of that
Oe in Species, supposine a the First Perm, and Cone and the semi-diameter ot its Base.
rche Ratio, here equal 1O 2.
a. ra. rra. rrra. rerra. ritica, _C..
For erery Term, burche first, arises From the mul-
siplicarion of che Ratio, Or Pome Porver oF ic, in-
10. Ic.
The Follomins Numbers are Conrinual PrOpOr-
rions decreasins, accordins co che Common Ratio,
oc 1.
128. 64. 32, 16. 8. 4,. 2. 16. 1
l iii ini
2. 4. 8. 16. (32, 64. 6c. i
As, Lupte
a, ra, rra in Or, 3. 6. 12.
oEADEFDM.
AIIO, I Three Ouanicies on each side are in
And From hence, Scurmius, in his Mathesis Fnu-
cleata, PrOp. 17. OF BOOK r. Schol. very brieAy de
monstrates chat Proposition ot Archimedes, That ibe