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IVI TITANICE, II
Note; What has been here done about whole Quantities id Simple Proportion, may be eafily performed in Fractional Quan: tities, and Surds, &c.
ab dc dt. For Inftance, If
; and if it be required to find the fourth Term, it will be do the Rectangle of the
fc Means ; which being divided by the firt Extream and will be
b) dd-ool ddccccc ddict come
the fourth Term. 2 fc abfc - abf Or if 6:vbd + bc:: Vodt bc: to a fourth Term. Then is, vbd+bcxv od tbc=bd+bc the Rectangle of the Means ; and 6) 6d-tbc (d to the fourth Term. That is, b:wbd too ::Vbd+bc:d + 6, &c.
Sect. 2. Of Duplicate and Triplicate Propotion, THE Proportions treated of in the last Section, are to be un
derstood when Lines are compared to Lines, and Superficies to Superficies ; or Solids to Solids, viz. when each is curdpared to that of it's like Kind, which is only called Simple Proportion."
But when Lines are compared to Superficies, or Lines are compared to Solids, such Comparisons are distinguished from the former, by the Names of Duplicate, and Triplicate, (&c.) Propor. tions ; so that Simple, Duplicate, and Triplicate, &c. Proporcions are to be understood in a different Sense from Simple, Double, Treble, sc. Proportions, which are only as 1, 2, 3, &c. to I; but those of Simple, Duplicate, Triplicate, &c. Proportions are those of a.aa.aaa . , &c. to 1. Or if the Simple Proportions
Sect. 3. How to turn Equations into Analogies.
TROM the first Section of this Chapter, it will be easy to con.
ceive how to turn or dissolve Equations into Analogies or Proportions. For if the Rectangle of two (or more) Quantities, be equal to the Rectangle of two (or more) Quantities; then are those four (or more) Quantities Proportional. By the 16 Eucl. 6. That is, if ab=id, then is a:0::d:b, or c:a::b:d, &c. From whence there arises this general Rule for turning Equations into Analogies.
: Cs":...* RULE
Divide either Side of the given Equation (if it can be done) into two such Parts, or Factors, as being multiplied together will produce that Side again; and make those two Parts the two Extrcams. Then divide the other side of the Équation (if it can be done) in the fame Manner as the first was, and let those two Parts or Factors be the two Means,
Of Substitution, and the Solution of Duadjatick
Sect. 2. The Solution of Duadratick Equations.
W H EN the Quantity fought is brought to an Equality with
those that are known, and is on one side of the Equation, in no more than two different Powers whose Indices are double one to another, those Equations are called Quadratick Equations Adfected ; and do fall under the Consideration of three Forms or