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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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Thus, luppole
0. 2. 4: 6. 8. 10. 12. 14. 16. 18, 2a
Thus, in che Collomins Series,
a. a  X. a  2. X. a l- 3X. a 1 4X. 5 ( 5X. a1
a -] 7X. a dd. a Gt. d TO.
The Sum of che Fxrremes 2 a = = 1OX, is equal
10 = a -) 5 X mulciplicd bY 2; thar is, the Double
oF the Midale Perm: AS ic is allo ro the Sum ot
any Trvo oF ehem equallr distane trom boch RX
tremes.
And ehis must erer be, because the last Perm
contains in icphe trst, and also che common Diste
rence super-added as otcen a5 che NTumber oE its
Place is distane from che Arst Perm: Bur, the ArA
Term hach no Aadirion of che Ditterence ac all;
and ac the Sccond Term hach one Difference Or
Rario more than the First, the Third one more
than the Sccond, (5c. Po the LaA lave onc, hach
one lels chan che LaH OE all; che L ast lare tWO,
one lels than che Las Care onc, (96. S0 thac rhe
Sum ot any Trvo oE chele equallr distane from che
Extremes, must be equal ro che Sum ot the Fx-
tremes; because One encreales as much 25 tHC
Ochex decreases.
tence ris plain,
COROL L A R TFS.
0. X. 2 X. 3X. 4 X. 5X. 6X. 7X 8N. 9X. XOX
20 che lase Term, mulriplied by er che Number
ot che Terms, gires 220; Thehalt ot irtiict, 11O,
is the sum of all the Terms.
For, By Corollary che First, 20, the Sum ot ehic
Fixrremes, mulciplied bY 5. 5 halt the Number Ot
Terms, produces Tr0, the Sum oE all che TerrS.
5. Ilence ehis very uletul Theorem may be de
duced, Thac ehie Sum oF all che Pemns in any such
Procression, beginnins From o, is Subduple oF the
Sum ot (omans Terms, all equal co the srearest
For ler che Erst Term beo, andche last x, and che
siven Number oF che Termsn; then (by PF L. 7
Wvill the Sum oF all che Terms oF the Progression
be Znx, Wtich is halF Ox subduple OF n X equal re
che Numiber oF s0 many Terms equal co e
grearest. 8. F. D.
sion, (i. c. aCcendine. bY the Cardinal Numbers
continued trom Vnirs ad Infinitum, is subduple the
Sum oF che same Number oF Terms, each oF
Which is equalrothe oreatest; or, on che contrarr.
this larrer Sum is double of che Former. By pre-
Y. That che Sum oP anx Dumber os Terrris in sixing a Cxpher Detore Vrier, this vould De bur 2
Tuch a Progresion may be trad, it the Sum of the Cale of the last Corollarr, che Sum oE the Pro-
Extremes be multiplied by hals the Number of the
Terms, Or halt chac Sum by che wwhole Numiber ob
rhe Perms.
2. F0 gain the Sum oE nerer lo many Onanti-
ties in rhis Progression, chere is noching neceslary tO
fression remainine Aillthe lame.
Buc chac ehis is erue in an infnite Series, begin-
mins from Onier, (For in a Finite Or Decerminare
one, the Proxorrion of the sum is alr 2ys less chan
double; tho ic alvays approaches coit, and comes
Po much che nearer by hou much srearer the
be siren bur che Extremes and the Number OF Scries i5) we Mall nou chus demonstrare:
Terms: S0 thac if by having the First Term and
Common Fxcess, one ould sec che Last, cvould
Stranselr disparch Ouestions in Progression.
3. Wilers sau Lem in Cuch a Progression may
behad caslr, by musciplvins ehe Number oF the
Terms, lessened by one, into the Common PXCelS.
and then 10 thac Produet addine the ArA PerM.
Thus it che last Term, in a Prozression ot 52
Places vere desired, utiere che Common Diterence
15 3, and che Aest PerM g:
Mulciply 5F by 3, ir produces 133; eo vhich
addino 5 che ArA Ferm, Vou have 158 cqual 1O
he last Perm in such a Prosression.
4. WVheretore iE che Progression begin wich a Cy
Pher, (wrhich is the most Nacural and Simple Pro-
pression; For when ic begins wich any other Perm,
T0 che Sum oF Three TermS, 1, 2, 3, 1. c. 6,
the Sum oF as man equal in Number co the
Sreatelt, 1. c. 9, hasthelame Proporrion a5 3t02;
but to the Sum OF Six TerS, 1, 2, 3, 4, 5, 6, 1. e
2T, the Sum of as many equal ro the grearest.
1. c. 36, has che Tame Proporrion 25 310 1 -] - 3
thav 15 3 FO 2 = 2 the Decreale beins =; bue rc
the Sum oF Tvelre Terms, wrhich mar befound
by Coroll. L. equal 78, che Sum ot s0 mans equal
co che greatest, viz. 244, has che lame Proportion,
(dividing boch des by 48) 45 3 10 128, 1. c. 310
1 -1 = - =, (For 24 matee in and the Remain-
der =5 is the Came 25 f) Thac i5, 25 3102 - -
theDecrement beins no J. Since cheretore, by
doublins che Number oF Terms. onvards, roull
find the Decrement r0 be =5, and lo onwards in
double Proportion; the Sum ot an Infinire Num-
ber oF such Terms, in Arichmerical ProgresTon.
equal to che grearest, will be tothe Sum of the
Procression, From r ad insinitum; 25 3 202 =
=E, (dc. Thac i6, bY Coroll. 1 and 2, 23
3 10 2 - =, Thac 15, 25 3EO 5 5, OF 25 2 F0 I.
D. E. D.
O
6. The Sum oF a Simple Arichmerical Progres.
eis in realier a Compound ot rvo Proeressions; onc
OF EqualS, 25 OF a, a, a, cn (90. aset l Oet 6
Arichnerical ProporrionalS, 25 0. X. 2 X. 3X. 4 X.
CSC.) chen I Cay the Sum of al the Terms will bc
equal ro halF che Produet oF che lac Term muiri-
Plicd by che hole Number ot ette letus. 10b)
rhe First Corollarr, the Su OL H EC LCMIS WII
2. Xhe Sum oe any Duplieate Aitumeriealero.
Ecito, Corsusca Tiom Onies ad insinitum, ls
lubetisse ot the Sum ot as manr Ferms equat 1b
be equal io ihe dum ol che lSCccies 
O halt ihe Dumber Ot Letirs.
EhC ElCaccnus do 1o ehe Leclidet et ECclto. Let
any Tuch linte Losreo IS Sreatet MMaM CHE SHD-
eriple Proportion, bur approaches nearer and near-
et 10 ic concimals, by hou much che laccher clie
Serieo