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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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3 =- a: LHal 1, IHC FIEH Lerm is Eo the Second )T6Th Place inthis PrOGreNTIOM, SincC 8 1- 8 = 7C
:: 25 che Sum oE all the Anrecedenrs is co rhe Sum Ifind, chac it I multiply 256 by ic PelF, or square
ot allche Consequenis: Wrheretore by mulciplica-
tion oE he Extreams and MeanS, Z4 = aa == r aZ
ray. Wheretore cranstere Za, and ic Will bc
aa
ra
29. Franspole ray
then Will r ay = a a = r aZ = Za. Divide eact
Pacc byr a = a, and Z will Be equal 1
Thae i5, Multiply the second and Last Terms toce-
sether, and from the Product substract the Square
the First Term; and then divide the Remainder b
the Difference betroeen the First and Second Term;
ond the Ouotient vill bethe sum of al the Terms.
Example in ehis Series 2. 4. 8. 16. 32. 64. 128.
ic, ic Wwill sire me 65536, Which is chersch place
and Cince 15 -)=7 = 23, iC Lmulripls chac Num
ber by 228, ic will produce 8388608, Wrhich isthe
24MH Place or Nail; (For cho Indices begin witho)
and cheretore char doubled, accordins ro he Con
dirion oF che Ouestion, will gire 16777216 Fac
things, wrtich reduced, i5 174761. 55. 4 d
From ertrich Propositions aboue Geometricat Pro-
porrionals, thele wondertul Corollaries may be de-
duced.
1. Fhac iis postible by chese Rules co collect an
infinice Series oF Proporrional Perms into one Sum
altho it is impossible co run over aI che Terms le
perately, becaule Intinire. Thus, in a conrinued
Series oF Fractions decreahns in a double ProxOr-
FiOn, 2, 23 2, 23, 32, 9c. ad Infinitum, il vou tate
them backards, ron mas justly recton a Cypher
Or 0; for the First Term; (for berween : and 0
chere mmax be an Intinice Number oF such Ferms
and the Infinire Sum ot chese Terms ill Be pre-
cisets equal r0 Vnier: For subrractine the first c
trom the last:, and che Remainder: beins divi-
ded by che Name ot che Realon sesten d by 1, char
is by i, erhich dirides noching; the Ouoriene : is
The Z oF allthe Lerms burthe last, mar be found che sum ot all che Terms, exceprins chie lat, (b)
very easily chus: From the Lastrake rhe First, and Rule 2.) and (0 che last = Beins added, the Sum oF
diride the Remainder by che Ratio le Nened by Oniey; aIl in chat Series will De T. Now it che last is
the Duocient is ehie Sum oF all the Terms burche ROF =, buc r, che Sum oF all Will necellarily bc2;
lact, a5 trill vers casils appear is rou multiply and
divide Algebraicallr: Which Rule is in some Ca-
ses more reads than the former, and cheretore ris
Tometimes reterred ro in che Collowins Consectaries
under cho Name ot Rule 3.C
P ROPOSTFTONV.
Llarins che Rario ot ene Lertus in ans beolle
trical Prosreition, To find any of the other Terms,
or 10 find an) term sousis. 3
and if 2 bc che last, che Sum oF all wilt be 4: In
a rord, it ill be alvars double the laA Perm.
2. And snce in this Case che Sunt oF all che pre-
cedent Terms is equal ro che last Perm, the one
beins subtracted From che orher, chere will remain
nochins; 1. c.
32 (9c. in In-
finitum, is = o; and al(0, x = = = =, (6. OF
2
= sn 9c. = 0.
3. In like manner the Sum oE Intinice Fractions
A5For Instance: In Progresion, whose Rario is decrealns in criple Rcalon in an Intinite Series
2, t0 find rhe 24th Perm; Or rO ansiver expediri-
ousy che Common Ouettion ot chie Price oF an
FPorse Cold at a F’arthins a Nail (oF his Stioe)
and doubled ererrtime.
Besin and double, a5 ris eatie to do, For 6 Or 8
rimes, phus,
0
2. 12. 5. 4
3. 16. 32, 64. 128. 256.
(3-
F =  53 - 51, (9c.) Will bc equal 10
Foril trom che last; (2gain in an inverted Order)
rou luberaet, che tirst o, and the Remainder  bC
dirided by che Name ot che Realon lessen d by V.
niry, chat is by 2, che Quoriene? will be che Sum
Ol al che Ancecedent Terms; and addine cortis
lalt5 Or 2, che Sum oC all will De L OF =
Buc had che Fractions decreasd from = in 2
Ouadruple, or From z in a Ouincuple Proporrion,
E5C. the Sum of che whole Series on d hare been
And then Over the Geometrical Dumbers place a laccordinelv = Or; and so any Series ot chis kind
Series oF Ariclimerical ones, besinnins wich 0, 25
is equal ro a Fraction, Whole Denominator is less
rou see: Thele are Indexes or Exponenes ot the by an Onite chan che Denominator of the last Fra
ocher, and Be erery where hou otien he Racio ction in that Series, (suppolins the Series to be
is multiplied into its lel ro produce anr parricular numbred backward, as betore.)
Term. Thus over 64, _ Ia Term, chere sands
che Index or Eexponenr 6; irhich sens chac 64 i5
4. Generally allo, any Insinite Series os tracti-
the schPorver of cheRario 2, (since the Series be. ons decrealins accordins ro the Proporrion ot the
Sins wich Oniry.) No che Addirion, and Substra- Denominator of che last Term, and havine a com-
etion ot the Inderes antvers ro the Mulciplicarion men Numieraror less by au Vnire phan rhe De-
and Division oF che Wumiberscher Rand oxer: For nominator of che last P et, (25 3- 3  3, 69c.
8, wrtich ische Index ot 256, the Pro- Or 3 -)
24 C9C. Or
190
duct oF. B into 32; ancl 7 = + 4 = 3, Which is che is équal eo Cniry, alter the Parrie vay as tie Se-
Index oF 8, ehC OnorienE OF F28 dirided by 16. ries, Coroll. L. Which may be comprehended under
Thisbeins che admirable Properer and vast Dse oF) ehis kind, and which may be demonstraced in al
these Indices, (wtich is the Reason and Founda ips parricular Cases by che Tame Method ve have
tion ot chelarhole Businels, oE the Losarithms, a5 hicherro made use ot, or allo barelr subsumed trom
roumay see under thar Word) ris very easie by
Coroll. 3. FOr Since
Cheir help to come ac anr Lerm, cho never (o re-( 10
- 3 - 1-
more citien az. For luprotins L moula have che) and l0 in Hhe rest.
(90. is equal
, 13c. Wwill be equal 10 = Or 1:
N
500 (254 = 254