358
IV - Powers, Exponentials, Logarithms, Trigonometric Functions
Newton did not know explicitly, or did not seek to discover, the general
formula for multiplying power series; it would not have cost him much trouble 18 and he would surely simply have shrugged his shoulders if one had made
such a remark to him. He was clearly capable, as he wrote to Leibniz in 1676,
of verifying that
(1 + x/2 - x 2 /8 + x 3 /16 - 5x 4 /128 + ... ) .
. (1 + x/2 - x 2 /8 + x 3 /16 - 5x 4 /128 + ... ) =
= (1 + x/2 - x 2 /8 + x 3 /16 - 5x 4 /128 + ... ) +
+ (x/2 + x 2 /4 - x 3 /16 + x 4 /32 - ... ) +
+ (_x 2 /8 - x 3 /16 + x 4 /64 - ... ) +
+ (x 3 /16 + x 4 /32 ... ) + (-5x 4 /128 - ... ) + ...
which reduces to 1 + x modulo terms of degree> 4. The reader who cares to
verify this modulo terms of degree > 8 may continue the calculation in the
same way; Newton once remarked that his brain had never worked so well as
when he was twenty ...
He was also capable, so as to confirm his formulae, of calculating (1+X)1/2
by extracting the square root of 1 + x as in commercial arithmetic; again the
analogy with decimal numbers, his inspiration for power series. The practice
of this sport 19 having now been lost, we shall not insist on this point. More
simply, if one postulates the existence of a formula
(1 + x)1/2 = 1 + alX + a2x2 + a3x3 + ... ,
one must have
l+x
(1 + alX + a2x 2 + a3x3 + ... ) (1 + alX + a2x2 + a3x3 + ... ) =
1 + 2alX + (a~ + 2a2) x 2 + (2ala2 + 2a3) x 3 + ... ,
whence al = 1/2, 1/4 + 2a2 = 0 i.e. a2 = -1/8, etc. This method allows
one to check that the first terms of (10) are correct, but it seems difficult to
18 The only obstacle would have been the absence of convenient notation. Indices,
the L:, the notation f(x), "bound" and "dummy" variables etc. were introduced
only later, hence in Newton's time all calculations were performed in a totally explicit way. In fact, it was Newton who was first to write the general formula for the
binomial coefficients, even for positive integer exponents; the "Pascal triangle",
which allows one to calculate them one-by-one, was not known before him. And
even Newton limited himself, later, to writing 8(8-1)(8-2) ... /1.2.3 ... without
specifying the last factors. The first modern notations, for example J f(x)dx, are
due to Leibniz, a philosopher and logician, and to the Bernoullis who were in
contact with him. One cannot overemphasise the rOle that well-chosen notation
has had, and continues to have, on the progress of mathematics.
19 John Wallis, in the books from which Newton learned analysis, is reputed to
have consecrated a night of insomnia to calculating in his head the square root
of a number of fifty digits to about fifty digits and to have dictated the result to
his secretary in the morning.
IV - Powers, Exponentials, Logarithms, Trigonometric Functions
Newton did not know explicitly, or did not seek to discover, the general
formula for multiplying power series; it would not have cost him much trouble 18 and he would surely simply have shrugged his shoulders if one had made
such a remark to him. He was clearly capable, as he wrote to Leibniz in 1676,
of verifying that
(1 + x/2 - x 2 /8 + x 3 /16 - 5x 4 /128 + ... ) .
. (1 + x/2 - x 2 /8 + x 3 /16 - 5x 4 /128 + ... ) =
= (1 + x/2 - x 2 /8 + x 3 /16 - 5x 4 /128 + ... ) +
+ (x/2 + x 2 /4 - x 3 /16 + x 4 /32 - ... ) +
+ (_x 2 /8 - x 3 /16 + x 4 /64 - ... ) +
+ (x 3 /16 + x 4 /32 ... ) + (-5x 4 /128 - ... ) + ...
which reduces to 1 + x modulo terms of degree> 4. The reader who cares to
verify this modulo terms of degree > 8 may continue the calculation in the
same way; Newton once remarked that his brain had never worked so well as
when he was twenty ...
He was also capable, so as to confirm his formulae, of calculating (1+X)1/2
by extracting the square root of 1 + x as in commercial arithmetic; again the
analogy with decimal numbers, his inspiration for power series. The practice
of this sport 19 having now been lost, we shall not insist on this point. More
simply, if one postulates the existence of a formula
(1 + x)1/2 = 1 + alX + a2x2 + a3x3 + ... ,
one must have
l+x
(1 + alX + a2x 2 + a3x3 + ... ) (1 + alX + a2x2 + a3x3 + ... ) =
1 + 2alX + (a~ + 2a2) x 2 + (2ala2 + 2a3) x 3 + ... ,
whence al = 1/2, 1/4 + 2a2 = 0 i.e. a2 = -1/8, etc. This method allows
one to check that the first terms of (10) are correct, but it seems difficult to
18 The only obstacle would have been the absence of convenient notation. Indices,
the L:, the notation f(x), "bound" and "dummy" variables etc. were introduced
only later, hence in Newton's time all calculations were performed in a totally explicit way. In fact, it was Newton who was first to write the general formula for the
binomial coefficients, even for positive integer exponents; the "Pascal triangle",
which allows one to calculate them one-by-one, was not known before him. And
even Newton limited himself, later, to writing 8(8-1)(8-2) ... /1.2.3 ... without
specifying the last factors. The first modern notations, for example J f(x)dx, are
due to Leibniz, a philosopher and logician, and to the Bernoullis who were in
contact with him. One cannot overemphasise the rOle that well-chosen notation
has had, and continues to have, on the progress of mathematics.
19 John Wallis, in the books from which Newton learned analysis, is reputed to
have consecrated a night of insomnia to calculating in his head the square root
of a number of fifty digits to about fifty digits and to have dictated the result to
his secretary in the morning.