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&c ad infinitum absolute term æquales arcus arithmetical mean binomial quantity binomial theorem cafe Cardan's co-efficients columns of terms complicated series compound quantity compound series consequently corresponding terms cube cube-root cubick equation cujus curva curvilined trapezia cylindricæ derived discourse equal equation qx equation x3 following even vertical following terms foregoing former fourth term fraction greater horizontal row hyperbola hyperbolical space ideo infinite series involve iooo less lesser root likewise logarithm lowest terms magnitude manner following marked multiplication number of terms obtained odd powers odd vertical columns prefixed quam quod ratio recta rectæ rectangulum residual quantity residual theorem resolving row of terms secants second term serieses sides sign f simple series sines square square-root subtracted suppose supposition tangents th power th root third term tion transcendental expression true value whole number whatsoever
Page 87 - ... to twelve. But if Briggs's firft twenty chiliads of logarithms be fuppofed made, as he has very carefully computed them to fourteen places, the firft ftep alone is capable to give the logarithm of any intermediate number true to all the places of thofe tables. After the fame manner may the difference of the faid two logarithms be very fitly applied to find the logarithms of prime numbers, having the logarithms of the two next numbers above and below them : for the difference of the ratio of a...
Page 591 - Saunderfon, in the fécond volume of his Algebra, in the chapter on the binomial theorem, where the reader will find a good explanation and illuftration of the faid celebrated theorem, by a variety of examples, both in the cafe of integral powers and in the cafe of roots, and other fractional powers, and even in the cafe of negative powers, and of powers that are both fractional and negative ; but no demonftration of it in any cafe, not even in that of integral and affirmative powers. FINIS.
Page 87 - If the curiosity of any gentleman that has leisure Would prompt him to undertake to do the logarithms of all prime numbers under 100,000 to 25 or 30 figures, I dare assure him that the facility of this method will invite him thereto; -nor can anything more easy be desired. And to encourage him, I here give the logarithms of the first prime numbers under 20 to 60 places.
Page 88 - I conceive was never till now perfected, without the confideration of the Hyperbola, which, in a matter, purely arithmetical, as this is, cannot fo properly be applied. But what follows, I think I may more juftly claim as, my own, viz. that the Logarithm of the...
Page 85 - Proportionals, and between i and 3 there will be 477 12 &c. of them ; which Numbers therefore are the Logarithms of the Rationes of i to 10, i to 2, and i to 3 ; and not fo properly to be called the Logaithms of 10, 2 and 3.
Page 91 - ... this be a matter purely arithmetical, nor properly demonftrable from the principles of geometry. Nor have I been obliged to have recourfe to the method of indivifibles, or the arithmetick of infinites, the whole being no other than an eafy corollary to Mr. Newton's General Theorem for forming Roots and Powers.
Page 85 - ... there be taken any infinity of mean proportionals, the infinitely little augment or decrement of the firft of thofe means from unity, will be a ratiuncula ; that is, the momentum or fluxion of the ratio of unity to the faid number. And feeing that in thefe continual proportionals all the...
Page 85 - Ratiunculas are to be fo underftood as in a continued Scale of Proportionals, infinite in Number between the two Terms of the Ratio ; which infinite Number of mean Proportionals is to that infinite Number of the like and equal...
Page 88 - The next prime Number is 23, which I will take for an example of the foregoing Doefcrinc ; and by the firft Rules...
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