366
IV - Powers, Exponentials, Logarithms, Trigonometric Functions
one might believe; 27 terms of the series suffice 23 , x is 1 or 2 divided by 10, 100
or 1000, so that the numbers to be calculated diminish as the degree increases
- the calculation, impeccably presented in a triangular form, occupying half
a page, is reproduced in the Mathematical Papers edited by Whiteside - and
the decimal fractions to be calculated are periodic: for x = 1/10 for example,
we have
X 17 /17 = 0.0 ... 080(5882352941176470)(5882352941176470)(58823
as the happy possessors of pocket calculators can verify if the latter are precise
enough, which is doubtful.
These calculations, though apparently futile, in fact lead to more interesting results, namely the log of 2, 3, 5, 7, 11, 13, 17 and 37. It is clear that his
series cannot provide them, but there are more intelligent methods, namely
the relations
2
1.2 x 1.2/0.8 x 0.9, 3 = 1.2 x 1.2 x 1.2/0.8 x 0.8 x 0.9,
5
2 x 2/0.8, 10 = 2 x 5, 11 = 10 x 1.1, etc.
found again in Euler. They furnish the desired log in stages, starting from
those already calculated, or from similar log: we have 37 = 1000 x 0.999/27,
whence
log 37 = 3. log 2 + 3. log 5 + 10g(1 - 1/1000) - 3. log 3,
the log of 1-1/1000 being calculated easily by the series. One might imagine
that, to check his calculations, Newton compared them to the available tables
of logarithms; but apparently not, which can be understood in view of his
prodigious advances on their authors, whom, in any case, he seems not to
have read. See Houzel, Analyse mathematique, pp. 79-82.
Newton did not imagine that at 23 years old he could have made discoveries capable of interesting the mathematicians; all his work remained in
his drawer. But in 1668, when he was again in Cambridge, there appeared
in London the Logarithmotechnia of the German Mercator (whose real name
was Kaufmann, "merchant") established in London. The series 10g(1 + x)
appears there, and even the series x 2 /2 - x 3 /2.3 + x 4 /3.4 - ... representing
the area of the function y = - log x contained between its asymptote and the
vertical at x (x < 1).
Newton found himself, for the first but not the last time, in the classical
situation of the scientist, a debutant moreover, who sees a colleague publish
results which he had found earlier but kept to himself; he had not shown them
even to Barrow, his patron, the Lucasian Professor of Mathematicks. Newton decided to speak to him, and was recommended to present his results in
23 Newton calculated separately the sum of the terms of even degree and of the
terms of odd degree; by addition and subtraction of the results he obtained
simultaneously log 1.1 and log 0.9.
IV - Powers, Exponentials, Logarithms, Trigonometric Functions
one might believe; 27 terms of the series suffice 23 , x is 1 or 2 divided by 10, 100
or 1000, so that the numbers to be calculated diminish as the degree increases
- the calculation, impeccably presented in a triangular form, occupying half
a page, is reproduced in the Mathematical Papers edited by Whiteside - and
the decimal fractions to be calculated are periodic: for x = 1/10 for example,
we have
X 17 /17 = 0.0 ... 080(5882352941176470)(5882352941176470)(58823
as the happy possessors of pocket calculators can verify if the latter are precise
enough, which is doubtful.
These calculations, though apparently futile, in fact lead to more interesting results, namely the log of 2, 3, 5, 7, 11, 13, 17 and 37. It is clear that his
series cannot provide them, but there are more intelligent methods, namely
the relations
2
1.2 x 1.2/0.8 x 0.9, 3 = 1.2 x 1.2 x 1.2/0.8 x 0.8 x 0.9,
5
2 x 2/0.8, 10 = 2 x 5, 11 = 10 x 1.1, etc.
found again in Euler. They furnish the desired log in stages, starting from
those already calculated, or from similar log: we have 37 = 1000 x 0.999/27,
whence
log 37 = 3. log 2 + 3. log 5 + 10g(1 - 1/1000) - 3. log 3,
the log of 1-1/1000 being calculated easily by the series. One might imagine
that, to check his calculations, Newton compared them to the available tables
of logarithms; but apparently not, which can be understood in view of his
prodigious advances on their authors, whom, in any case, he seems not to
have read. See Houzel, Analyse mathematique, pp. 79-82.
Newton did not imagine that at 23 years old he could have made discoveries capable of interesting the mathematicians; all his work remained in
his drawer. But in 1668, when he was again in Cambridge, there appeared
in London the Logarithmotechnia of the German Mercator (whose real name
was Kaufmann, "merchant") established in London. The series 10g(1 + x)
appears there, and even the series x 2 /2 - x 3 /2.3 + x 4 /3.4 - ... representing
the area of the function y = - log x contained between its asymptote and the
vertical at x (x < 1).
Newton found himself, for the first but not the last time, in the classical
situation of the scientist, a debutant moreover, who sees a colleague publish
results which he had found earlier but kept to himself; he had not shown them
even to Barrow, his patron, the Lucasian Professor of Mathematicks. Newton decided to speak to him, and was recommended to present his results in
23 Newton calculated separately the sum of the terms of even degree and of the
terms of odd degree; by addition and subtraction of the results he obtained
simultaneously log 1.1 and log 0.9.