646. AN CHAP V. Of the Arithmetick of Infinites. Demonftration. N Infiniteffimal is that which is conceived to be infinitely lefs than any affignable Quantity, as a Grain of Sand in Comparison with the whole Globe of the Earth; a Moment of Time in refpect of a Million of Ages, &c. COROLLARY XVI. 647. Every Infiniteffimal is comparatively as nothing in refpect of the Quantity to which it is an Infiniteffimal. COROLLARY XVII. 648. Hence whatever Quantities do only differ by an Infiniteffimal are to be looked upon as equal, and confequently may be fubftituted one for another. COROLLARY XVIII. 649. Hence alfo every inferior Power of an infinite Quantity (if I may be allowed the Expreffion) is an Infinitesfimal in respect of its fuperior one, as being infinitely lefs than the other; i. e. the Square of an Infinite is an Infinitefimal in refpect of the Cube, the Cube in refpect of the Biquadrate, &c. therefore every inferior Power is to be looked upon as nothing in respect of its fuperior one. COROLLARY XIX. 650. Whence, if a Series of Laterals 1, 2, 3, 4, 5, &c. be continued ad Infinitum; fo that n the greatest Term and Number of Terms be infinite; then, rejecting all the Infiniteffimals by the laft, the Theorems A (In. 2 213.) will become - ; B= n2 2n3 +3n2+n 2 213 will become B- or C n^ —-2n3 + n2 6 3 4 4 12 12 651. Whence laftly have we this general Theorem, That the Sum of every Series of Homologous Powers, whofe common Exponent is m, and whofe Roots are the natural Numbers 1, 2, 3, 4, 5, &c. ad infinitum, is to as many equal to the greatest as I to m+1. In particular, If m=1, the Sum of the Series of Firft Powers or Roots A is to as many equal to the greatest, as 1 to 2: If m=2, the Sum of the Series of Squares B is to as many equal to the greatest as 1 to 3: If m=3, the Sum of the Series of Cubes C is to as many equal to the greatest as 1 to 4. If m=5, the Sum of the Series of Biquadrates D is to as many equal to the greatest as 1 to 5, &c. ad infinitum. SCHOLIUM IX. 652. And by the fame Manner many other Theorems may be raised from In. 632, 635 and 641. The Inventor of the Arithmetick of Infinites was the great Dr. Wallis, Savilian Profeffor of Geometry in Oxford; the use of which is generally in the Bufinefs of Geometry. But fince the admirable Invention of Fluxions, it is for the moft Part laid afide. The End of the Fifth PART. Bb ARITH ARITHMETICAL INSTITUTIONS. PART VI. The APPLICATION of SPECIES ALGORISM to the EFFECTION OF INDETERMINATE PROBLEMS. 653. T CHAP. I. Of Single LATERAL EQUALITIES. PROBLEM I. O find two Integers a, e, whofe Sum added to their Product is equal to a given Integer n. Therefore for e must be affumed fome Number, which being added to Unity will divide n+1 without a Remainder. Ex. gr. If n=19, then e which a 3; ore may 9, according to which a=1. PROBLEM II. 654. To find two Integers a, e, whofe Difference added to their Product is equal to a given Number 1. Therefore for e muft be affumed fome Number which being added to Unity will divide n+1 without a Remainder. Ex. gr. If n=28, then e may =2, according to which a=- +1=10, or e may 8, according to which a=4. SCHOLIUM I. 55. In the Effection then of the two foregoing Problems, it is plain that n in the former, and 1 in the latter muft always be compofit Numaffumed any of their Aliquot Parts. bers; and for e+-1 may be PROBLEM III. 656. To find two Integers a, e, whofe Sum is equal to their Product lefs the latter. Whence 2a=. 22 = 2 + ==== Therefore for e here can be affumed no Integer but 2, and accordingly 657. To find two Numbers, a the greater, the leffer, whofe Sum is equal to the Difference of their Squares. Effection. |