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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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R
eR OPOSTTION H
IF Four Ouantities are proporrional, cithex con-
cinualle or dilereetly, che Produet oE che fixtremes
is equalto rhe Product oF the MeanS.
For il an ray rray rera, in Or 2. 4. 8. 16. .
ris plain che Produet oF the Extremes and oF che
Vleans, is rhe same Ouanrity rrraa OF 32.
In Discreer Proporionals, let a: ra:: b: rb;
that 15, 2: 4 : : 15 : 30. The Product OF the FX-
treams ard of che Means can here beno orhet char
ab = 60.
On Wohich Proporrion is grounded che Colden
Rule, Or Rule of Threc in Arithmerick; (0 called,
because havins Threc NTumbers, (25 2. 4. 25.) ic
ceaches hion to find an VInKHOWn FOUrtH PFOPOT-
tional: For altho this Fourth be unknon, yec ic
Produet bV 2, cheFirst Perm, is knomn, becaule ris
the lame wich the Product oF che McanS + and 15.
Wheretore che Rule directo us ro multiply che
Third by che Sccond, chac rou may chereby ob-
tain the Product oF che Extremes; wtrich dirided
by onc oF the Fxrremes, viz. the First, must necds
sire cheorher, that is, the Fourch Toughes
COROLEREES
Etence vis plain, Thar iC. Lvo Products arisine
From che Nulriplication ot any LWo Pairs oF Onan-
ecies, are equal; chose Four Ouantiries will Be ac
cast diserecels proportional.
PROPOSITION HI
IE chere are never somany Continual Proponion-
als, the Produet oF the Lxrremes is equal eo the
Produst oF any Two OF the MeanS Thar arc equa-
Iy distant trom che Farremes, 25 al(0 to ehe Square
ot che Mean or Midale Term, il che. Number ot
the Permsbe odd. le
Thus in etis ecogresion,
2 4 8 16 32, 64 128.
a. ra. rra. rrras rrra. rrerra. rrerrra. c.
IEs plain che Product dE tHC PXIreMeS, and oE any
Tuo Terms equally distane from them, and che
Square oF ihe Midalo Term, must alwarsibe
6aa.
As Etie las Tertm bue one, is muliiplied into 2
DeEreC OF POWer OF the Ratio lels By One than chC
last; (0 ehie Second Term is mulciplied into one
more than ehe First: And cheretore che Rectangle
OF thie Fxtremes mus RilI De che Tame Ouantity
wich chat of the Product of any Tvo Terms c.
quallu distant from chat; and chis also equal ro che
Square oF the Midale Lerm, it the Number ot
ehem be odd.
P R O POSF Y LO NmA
Flaviris rhe First and Last Ferms, and ete Rario;
Tofind the Sum os all the Terms in any Ceometrical
Proeression, MI. Ousbtred siveschis Mertod.
Let a beche sirst Perm, andy the last; etien let
all the antecedents, and Z = a = 3Il che
Consequents. Theretore, by Compolition oF Pro-
porcion, ic plainls CoHlows, Thar a. ra :  Z 7
LeF F F (OF rb) be a Mean Proporional DC
Eveen the Side oF the Cone B C, (OF rrb;) atic
ihie Radius oF rhe Basc CD, (Or b;) and let
etiere be an equal Number oF Peripheries, rra
ra, and a artiverine 10 un equal Nuiber OF Ra
dii in che lame proxorrion; then Wvill halt che firte
Radius B C, (Ox rr B) mulciplied inco the Peri-
phery a, (thac is = rr a b) be equal ro halF che
eroaut ot uhe midase Line imco chemidale Peri
pherr, chat is (as Betore) halt rrab: The FOF
irier ot wrhich is equal co che surtace oF the Given
Cone, (by the Ach Cons. OF his De. 18.) and ehie
latter ro the Area oF che Circle, Wio Lacius I5
the Mean Proporion EF, (by rhe 2dC09. et tii
Def. 15.) 8. F. D.
Ana Trom hence also Rous naturally chis orhet
Proxotirion, That the Sursace B sc Lo, Bals L lL b,
os s Base hals 2b; ac the Side ef the Cone P F
s to obe Radius es che Pasc b. F0r
Errab: ab:: rrb: b.