SCHOLIUM II. Here the Quotient is an Infinite Geometrical Series, whofe d first Term is or a=1db, and common Multiplier is a-ibi + eee aaaa T&c. d or 402. The Quotients of the four laft Examples to the foregoing Problem are ftiled Infinitinomials; where it may be obferved, that if d in the fecond Example be put for the Numerator, and a for the Denominator lefs 1 of any Vulgar Vulgar Fraction, and 61, we will have all Vulgar Fractions, except ¦, expreffed by an Infinite Series, thus : In like manner, if b=1 d=1, and a≈ any Number greater than Unity in ftant Law for expreffing any Vulgar Fraction by an Infinite Series, whofe Numerator is Unity, and Denominator a less 1. And, if in the fame Example, b ftill equal Unity, and d be put for the Numerator of a Vulgar Fraction, and a for the Denominator increafed by Unity, then have we another univerfal Law for expreffing Vulgar Fractions, as follows, COROLLARY XVI. &c. 403. Whence it appears, that if a be put univerfally for any Integer, Which gives an univerfal Law for expreffing all Integers by an Infinite Series an infinite Number of Ways. Otherwife Otherwise any Integer may be put into an infinite Series of the fame Value, according to the laft Example of the foregoing Problem, by making a=2, and b=1, and d= the given Integer; thus, CHA P. III. Of the Compofition and Refolution of Powers. PROBLEM III. ROM the Binomial Root be (i. e. bte or be) to raise any Effection. 1. Put m for the Exponent of the required Power, by which Means the m m Power of b±e will be expreffed by b±e 2. Set down the Geometrical Series b bms bang bang brand bring &c. till the laft Term be lo=1. 3. Multiply each Term of the foregoing Series into each Term of this following one, viz. into c°1 e e2 e3 64 es &c. and the Series that refults will be bm bm-re bm-2e2 6m-360 bm-aca bm-ses &c. till the laft Term be bem=em 4. Connect all the Terms with the Sign+, if be was the Root: And with the Signs-+-+-+&c. interchangeably, if b-e was the Root; thus bbme+bm2 2±bm-8¿ 3 +6m-met±6m-ses &c. 5. For the Coefficient of the first Term put 1, of the second Term 1x m- -OM-I =m, of the third Term 1x -X- =p, of the fourth Term 1x2 X I M-I I 2 M-2 X q,of the fifthTerm 1x X -X X Fr, of the fixth Term 3 m m-Im-2 m-3 ̧m bm-33.x mm-I M- 2 m M-1 M-2 m-3 X X I X 4 5 6 X X X. 2 3 4 5 m—3 m―4 m—56m-6,6 &c. = bm±mbm=1e + pbm—2 e2 ± b±e = b2±7b°e +21b3e2±35b*e3+35b3e2±2xb1e3+7be±e? b±e = b2±8b7e+28be2±56b3é3 +70b*c*±56b3é3¬†28b2e° ±8be¶+e3 b±e=b3±9b3e +36be2±84be3+126b3e*±126l *e3 +84l3c6±36b1e•·+9be®±e® Otherwise the Powers of b±e may be raised by Multiplication, according to (In. 400.) as follows; COROLLARY XVI. 405. The Square of the Sum of any two Quantities is equal to the Sum of the Squares of the faid two Quantities, together with twice the Product which arifes by multiplying one into the other. The Cube of the Sum of any two Quantities is equal to the Sum of their Cubes, together with thrice the Product which arifes by multiplying the leffer into the Square of the greater, and thrice the Product which arifes by multiplying the greater into the Square of the leffer, &c. as in the Diagrams above. COROLLARY XVII. 406. The Square of the Difference between any two Quantities is equal to the Sum of the Squares of the faid Quantities lefs twice the Product which is made by multiplying one into the other. The Cube of the Difference between any two Quantities is equal to the Difference of their Cubes made lefs by the Difference between thrice the Product of the leffer into the Square of the greater, and thrice the Product of the greater into the Square of the lef fer. And fo for higher Powers, as in the foregoing Diagrams. COROLLARY XVIII. 407. Every Power raised from a Binomial Root confists of one Term more than its Exponent. COROLLARY XIX. 408. In the Scale of Powers whofe Root is b±e, the Coefficients of each (which by the celebrated Mr. Oughtred are ftiled their Uncia) proceed thus. ft, The Uncia of every firft Term is Unity. 2dly, The Uncia of every fecond Term is the Exponent of the Power m. 3dly, The Uncia of every third Term is a Triangular Number, whofe Root or Side is m-1. 4thly, The Uncia of every fourth Term is a firft Pyramidal Triangular, whofe Root is m-2. 5thly, The Uncia of every fifth Term is a fecond Pyramidal Triangular, whose Root is m-3. 6thly, The Uncia of every fixth Term is a third Pyramidal Triangular, whofe Root is m-4, &c. From whence is grounded the last Precept in the foregoing Effection. COROLLARY XX. 409. The Uncia of the firft and laft Terms are the fame, alfo the Uncia of the fecond and laft but one, of the third and laft but two, of the fourth and last but three, &c. COROLLARY XXI, 410. The Sum of all the Uncia of any Power raifed from a Binomial is equal to the Homologous Power of 2 or 1+1. Thus, the Sum of the |