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2. The Indices last found, are the two Extreams of four Num. bers in Arithmetical Progression, the lesser Mean being the Index of the firft Letter of the third Word ; and the greater Mean is the Index of the fourth and laft Letter of the firft Word. Viz. 5.7.9. IT are the four Terms in Arithmetical Progression. Whence it appears, that G (whose lodex is 7) is the first Letter of the third Word; and that i (whose Index is 9) is the fourth or laft Letter of the first Word; which being placed down, will stand thus,

* * li. *e* * * G **** 3. The second Letter of the third Word is the same with the third Letter of the firft Word; and the fifth Letter of the third Word is the same with the last Letter of the first Word: whence the Letters will stand thus,

* * li. *e * * * G * * * 1 *. 4. The Sum of the Squares of the Indices of the first and second Letters of the firft Word is 520, and the Product of the same Indices is seven Ninths of the Square of the greater Index, which is the Index of the said first Letter.

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7. The Sum of the Indices of the fourth Letter of the third Word, and the fixth or last Letter of the fame Word, being added to their Product is 35'; and the Difference of their Squares is 288 ; tbe Index of the last Letter being the least.

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This last Equation being resolved according to the Method which shall be Thewed in the next Chapter, it will be a = 17 it's Letter ; and from the 4th Step e=3= =1, the Index of

a+I the Letter a. Then these two Letters being placed according to the Data above, are all that are required by the Enigma to come pleat these Words.

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снАР. х. The Solution of Adfedted Equations in Numbers. BEFORE we proceed to the Solution of Adfected Equations, D it may not be amiss to shew the Investigation (or Invention) of those Theorems or Rules for extracting the Roots of Simple Powers, made use of in Chapter 11. Part 1. I shall here make Choice of the fame Letters to represent the Numbers both given and sought in my Compendium of Algebra.

PG, always denote the given Resolvend.

Sany Number taken as near the true Root as Viz. Let' = I may be, whether it be greater or less.

S the unknown Part of the Root fought by

l which r is to be either increased or decreased. Then if r be any Number less than the true Root, it will be ate the Root sought. But if r be taken greater than the true Root, it will then bere the Root sought. And put D for the Dividend that is produced from G, after it is lefsened and divided by r, &c. (into the Co-efficients of Adfected Equations) according as the Nature of the Root requires. These Things being premised, we may proceed to raising the Theorems.

SECT. 1. · I. FOR the Square Root, viz. a a=G. Quære a.

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Which gives this Theorem

stic The Arithmetical Operations of both these Theorems, you have in the Examples of Section 2. Page 126, to which I refer

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the Learner, supposing him by this Time to understand them without any more Words than what is there expreft.

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Which gives this Theorem ar?

By this Theorem the third Example in Case 2. Page 133, is performed.

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