## The Doctrine of Permutations and Combinations: Being an Essential and Fundamental Part of the Doctrine of Chances; as it is Delivered by Mr. James Bernoulli, in His Excellent Treatise on the Doctrine of Chances, Intitled, Ars Conjectandi, and by the Celebrated Dr. John Wallis, of Oxford, in a Tract Intitled from the Subject, and Published at the End of His Treatise on Algebra: in the Former of which Tracts is Contained a Demonstration of Sir Isaac Newton's Famous Binomial Theorem, in the Cases of Integral Powers, and of the Reciprocals of Integral Powers. Together with Some Useful Mathematical Tracts. Published by Francis Maseres |

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### Common terms and phrases

Aliquot binomial quantity Binomial Theorem cafe column of terms compound quantity consequently Coroll Cube cube-root cyphers denoted divided exponent expression fame numbers fifth power fifth root figurate numbers following terms foregoing table former fourth fraction given number greater horizontal row Isaac Newton James Bernoulli Lagny Lagny's last term lemma less letters Multiplied natural numbers Newton's method Number of Divisors number of permutations number of terms number of things numeral co-efficients numerorum obtained odd numbers preceeding Prime Numbers proposed number quadratick equation quæ quaternions quotient Raphson's ratio right-angled triangle row of terms SCHOLIUM second near value second term sides significant terms sixth square numbers square-root subtracted summæ table of combinations termini terminorum terms all equal Theorem things exposed third three numbers tion Tract true value vertical column virgo whole numbers

### Popular passages

Page 36 - ... to form on all the subjects we reflect upon, whether they relate to the knowledge of Nature, or the merits and motives of human actions. It must therefore be acknowledged that that art which affords a cure to this weakness or defect of our understandings, and teaches us to enumerate all the possible ways in which a given number of things may be mixed and combined together, and that we may be certain that we have not omitted any one arrangement of them that can lead to the object of our enquiry,...

Page 417 - Geometry ; where it is proved, that the fquare of the hypothenufe, or longeft fide of a right-angled triangle, is equal to the fum of the fquares of the bafe and perpendicular, or the other two fides.

Page 568 - ... to be performed : and, as to the negative roots of an equation, they are in truth the real and pofitive roots of another equation confiding of the fame terms as the firft equation, but with different figns + and — prefixed to fome of them ; fo that, when writers of Algebra...

Page 593 - For this root will ahvays be lefs than the leaft root of the original equation, if it really has (as it 'appears to have,) more than one real and affirmative root ; or it will be lefs than the only root of the original equation, if (notwithftanding the appearances to the contrary,) it really has but one root. When the root of this fécond, or curtailed, equation, has been difcovered, it may be called...

Page 589 - Raphfon's way of proceeding we have only to multiply x, or 2.0946 — w, into the very fimple co-efficient 2. So that, upon the whole, the difference of the labour of computation in the two methods is not very confiderable, though it is rather lefs in Sir Ifaac Newton's method than in Mr. Raphfon's. But in point of fimplicity of conception Mr. Raphfon's method feems much fuperiour to Sir Ifaac's, becaufe it never lofes fight of the original equation x3 — ^x == 5, which is to be refolved.

Page 35 - Nature and in the actions of man, and which constitutes the greatest part of the beauty of the Universe, is owing to the multitude of different ways in which its several parts are mixed with, or placed near, each other.

Page 49 - Lex , Rex , Grex , Res , Spes, Jus , Thus , Sal , Sol (bona), Lux, Laus. Mars, Mors, Sors, Lis, Vis, Styx, Pus, Nox, Fex (mala) Crux, Fraus.

Page 36 - Nor is this art or doctrine to be considered merely a branch of the mathematical sciences, for it has a- relation to almost every species of useful knowledge that the mind of man can be employed upon. It proceeds, indeed, upon mathematical principles in calculating the number of the combinations of the things proposed ; but by the conclusions that are obtained by it the sagacity of the natural philosopher, the exactness of the historian...

Page 565 - ... with lefs trouble than the particular and accurate methods above-mentioned, which have been invented for that purpofe. So that thefe methods of refolving equations by approximation ought to be confidered as of the higheft utility, and as being abfolutely neceflary to the completion of the Doctrine of the Refolution of Algebräick Equations, which is the moil important branch of the Science of Algebra.