ADDENDA ET CORRIGENDA

ADDENDA ET CORRIGENDA.

FLUENTS, VOL. IV. No. 5. For $\backslash int\; y\; dz$ read $\backslash int\; z\; dy$

No. 203. For $\backslash frac\{x^2\; \backslash sqrt\{y\}\}\{5b\}$, read $\backslash frac\{y^\{\backslash frac\{5\}\{2\}\}\}\{5b\}$

After No. 546. for $\backslash int\; Y\; dz$, read $\backslash int\; Z\; dY$; in the next line read ... (n. 5), taking the $dY$ of this theorem $=\; \backslash text\{hl\}^n\; \backslash frac\{dx\}\{x\}$, $dX\; =\; \backslash frac\{dx\}\{x\}$, and $Z\; =\; yx$, we have

After No. 555. for $\backslash int\; Y\; dz$, read $\backslash int\; Z\; dY\; =\; \backslash frac\{dY\}\{dX\}\; \backslash int\; Z\; dX$

..., we may put $dX\; =\; dx$, and either $ydX\; =$

$dY$, and $a^x\; =\; Z$, or $a^x\; dx\; =\; dY$, and $y\; =\; Z$; and

After No. 569. for $\backslash int\; Y\; dz$, read $\backslash int\; Z\; dY\; \backslash dots\; dY\; =$

INTERPOLATION. Few disquisitions in modern mathematics have greater practical utility than those concerning the extension and interpolation of the terms of numerical progressions. But the modes of solution generally proposed, however ingenious and refined, are often too complex for ordinary purposes. We therefore avail ourselves of this opportunity to point out a procedure of extreme simplicity and most ready application. It will embrace any system of numbers, but seems peculiarly fitted for the computation of logarithms.

Napier first published, in 1614, his canon Mirificus Logarithmorum, comprised in a very thin and small quarto volume, exhibiting, as far as seven places, the logarithms only of the sines and tangents to every minute of the quadrant; those of the ordinary numbers being left to be deduced from the nearest sines or tangents. In the summer following, Briggs, then Professor of Geometry in Gresham College at Lon-

VOL. VI. PART II.

$e^\{-xx\}\; dx$, $dX\; =\; d(-xx)\; =\; -2x\; dx$, and $Z\; =\; \backslash frac\{1\}\{x\}$

we have, 570. $\backslash int\; e^\{-xx\}\; dx\; =\; -e^\{-xx\}(x)\; +$

No. 571. A similar correction is required for $\backslash int\; \backslash frac\{dy\}\{\backslash sqrt\{\backslash text\{hl\}^n\; y\}\}$

which becomes $=\; \backslash sqrt\{\backslash frac\{y\}\{\backslash text\{hl\}^n\; y\}\}\; \backslash left(\; 1\; -\; \backslash frac\{1\}\{2(\backslash text\{hl\}^n\; y)\}\; +$

See Méc. Cél. X. p. 253.

No. 572. read $\backslash int\; e^\{x^m\}\; dx\; =\; e^\{x^m\}\; x\; -\; e^\{x^m\}\; \backslash frac\{m\}\{m+1\}\; x^\{m+1\}\; +\; \backslash dots$

Add—Some other similar fluents of logarithmic quantities may be found in the Journal of the Royal Institution for October 1823.

don, who was enchanted by that noble discovery, paid a visit to its illustrious author at Merchiston, in the vicinity of Edinburgh. During an agreeable stay of a month, it was concerted between them, to change the natural system of logarithms into another of a more artificial form, but adapted to our denary scale of notation, the labour of the calculation, however, being devolved upon Briggs, as younger and enjoying robust health. Next season, Briggs performed a second journey to Edinburgh, and showed to Napier a table which he had computed of the new logarithms for the first chiliad or thousand of the series of natural numbers. But the great inventor was now fast declining in years, and expired on the 3d of April 1618. Briggs employing most skilfully all the abbreviations which ingenuity could devise, prosecuted his most arduous task with such vigour and active perseverance, as to compute, in the space of seven years, to fourteen places of figures, not only the logarithms of

Interpolation. the sines and tangents of every degree and centesimal minute of the quadrant, but also the logarithms of thirty chiliads of numbers. These tables were printed, in 1624, in a folio volume, entitled Arithmetica Logarithmica. In his anxiety, however, to bring out this stupendous work, Briggs contented himself with calculating the logarithms from unit to 20,000, and from 90,000 to 100,000, leaving the computation of the seventy intervening chiliads to be afterwards supplied. The very ingenious author annexed a full explanation of his mode of framing the tables, grounded chiefly on the consideration of differences, and he thus traced the first steps of that important theory. But he likewise gave instructions for the reader filling up of the intermediate logarithms, abridging the toil of calculation no doubt, yet detailing a procedure sufficiently irksome and complicated. "To encourage some skilful persons to perform this task, he offered to furnish them with paper he had by him, ready prepared, and divided into columns proper for that purpose, as likewise to inform them at what part to begin, that they might not interfere one with another; and promised, when the whole was finished, to endeavour to procure a new edition of the work so completed." This slender boon seems not to have tempted the mathematicians in England. In Holland, however, Adrian Vlacy, chancing to meet two years afterwards with a copy of the work, and prompted only by his patriotic zeal, had the resolution to revise and compute the whole canon, reducing it to ten places of figures, which he printed in folio at Gouda, as early as 1628. The same able calculator, only five years afterwards, published, likewise in folio, at Gouda, a very extensive system of logarithmic sines and tangents, to every ten seconds of the quadrant, having restored the sexagesimal subdivision, which Briggs had partly changed into a centesimal one. These two volumes may be deemed a precious thesaurus of logarithms, from which succeeding compilers have drawn very liberally. They form the basis of the Tables published by Vega at Leipsic in 1794, which are esteemed the best and completest now extant.

But though the Tables of Vlacy, carried only to ten places of figures, are sufficiently accurate for every ordinary purpose, and even for the most delicate calculations in astronomy, yet many persons have often regretted that the original system of Briggs was never completed. The celebrated Legendre has employed that table, imperfect as it is, in some of his most refined numerical investigations. It is well known that Mr Baron Maseres devoted a considerable portion of his time, and of his fortune, to the republication of the works of the early writers on logarithms. In the course of this extensive undertaking, he entertained some thoughts of giving a new edition of the Arithmetica Logarithmica, and expressed an earnest wish that the vacant chiliads were filled up. To promote the liberal designs of the Baron, the Author of this Article was induced to bestow some reflection on the subject, and a very simple mode occurred to him, which would have reduced the labour of computing those logarithms to little more than the trouble of mere transcription. But

Interpolation. the project of completing the canon was deferred for a time, and afterwards gradually forgotten. The method of interpolation then proposed seems, however, to deserve notice on account of its great simplicity, and its ready application, not only to the immediate object, but to other questions of a similar nature. We shall, therefore, now state the principle, and illustrate its application by a few examples.

The square root of the quantity $a^2+1$ is evidently expressed by the continued fraction $a+1$

If two terms only of the fractional part be taken, the expression will become $\backslash sqrt\{a^2+1\}\; =\; \backslash frac\{4a^3+3a\}\{4a^2+1\}$, and con-

sequently $\backslash sqrt\{\backslash left(\backslash frac\{a^2+1\}\{a^2\}\backslash right)\}$ or $\backslash sqrt\{\backslash left(\backslash frac\{2a^2+2\}\{2a^2\}\backslash right)\}\; =\; \backslash frac\{4a^2+3\}\{4a^2+1\}$, a very near approximation. Put $b=2a^2$, and by sub-

stitution $\backslash sqrt\{\backslash left(\backslash frac\{b+2\}\{b\}\backslash right)\}\; =\; \backslash frac\{2b+3\}\{2b+1\}$; wherefore

$\backslash frac\{1\}\{2\}\; \backslash log.\; \backslash frac\{b+2\}\{b\}\; =\; \backslash log.\; \backslash frac\{2b+3\}\{2b+1\}$. Hence half the alternate

differences of the logarithms of the series $b,\; b+1,\; b+2,\; \backslash dots$ added to the logarithm of $2b+1$, must give the logarithm of $2b+3$. By this simple process then, any table of logarithms is carried to double its actual extent through all the odd numbers, those of the even ones being found by the mere addition of the logarithm of 2.

To find the limits of approximation, let three terms of the fractional series be taken, and $\backslash sqrt\{\backslash left(\backslash frac\{a^2+1\}\{a^2\}\backslash right)\}\; =$

and, therefore, since this last quantity exceeds unit only by a very minute difference, $\backslash frac\{1\}\{2\}\; \backslash log.\; \backslash frac\{b+2\}\{b\}\; =$

modulus of the system. If the number $2b+3$ or $2b+1$ be expressed by $N$, this small correction will amount

the correction on the first approximation will only reach unit in the last figure for the numbers un-

Interpolation. der 206 in tables of seven places, for those under 2055 in the tables of ten places, and for the numbers under 44286 in tables extending to fourteen places.

The corrections needed in Briggs's Tables will therefore correspond to these limits,

Suppose it were required to find the logarithms of the odd numbers above 300, to seven places of decimals. Assuming the logarithms of the series of half those numbers, let their alternate differences be taken, and these again bisected.

Hence the logarithms of the doubles are formed by the mere addition of these halves.

In this manner, the operation may be continued; but, to prevent any accumulation of errors, the logarithms of the composite numbers should serve as standards, being formed by the addition of the logarithms of their several factors. The logarithms of the intermediate even numbers, 302, 304, 306, 308, and 310, are easily determined by adding .3010300 to the logarithms of 151, 152, 153, 154, and 155.

To extend the process a little farther, let Vlaccq's logarithms be computed for the numbers above 4000.

Whence are derived,

Again, to compute the logarithms in Briggs's Canon.

The additive parts here consist of those differences diminished by 11 and the last one by 10, since the numbers now approach to the limit 20557.

The first mode of interpolating, thus derived from the nature of logarithms, and so commodious for their computation, might likewise be deduced from general considerations. Let $A,\; B,\; C,\; D,\; E$, &c. represent any series of numbers. If they advance regularly and slowly, their first differences, $B-A,\; C-B,\; D-C,\; E-D$, &c. may be viewed as constituting an arithmetical progression. Wherefore the sum of the extremes will be equal to that of the mean terms, or $(E-D)\; +\; (B-A)\; =\; (D-C)\; +\; (C-B)$, that is, $E-A\; =\; 2D-2B$, and therefore $D-B\; =\; \backslash frac\{E-A\}\{2\}$, whence $D\; =$

$B\; +\; \backslash frac\{E-A\}\{2\}$. Applying this to the logarithms of eight places of figures, let $A,\; B,\; C,\; D,\; E$, &c. represent

$-2.39794001)\; =\; 2.79156799$; the halves of the alternate differences of the logarithms of 250, 251, 252, 253, &c. being thus taken, as before, to compose by their additions the logarithms of the odd numbers 503, 505, 507, &c.

But since $E-A=2D-2B$, it follows that $E=A+2(D-B)$. Wherefore, in any series, the fifth term will be found nearly, by adding to the first term the double the difference between the second and fourth terms. In this way the tables of natural sines, tangents, and secants, could easily be framed. Thus, the logarithmic sines of the successive arcs, $50^\backslash circ$, $50^\backslash circ\; 1\text{'}$, $50^\backslash circ\; 2\text{'}$, and $50^\backslash circ\; 3\text{'}$ being from Vlacy's Tables, to find the logarithmic sine of $50^\backslash circ\; 4\text{'}$.

Here, passing over the middle term, the difference between the logarithmic sines of $50^\backslash circ\; 1\text{'}$ and of $50^\backslash circ\; 3\text{'}$ is doubled, and added to that of $50^\backslash circ$, to form the logarithmic sine of $50^\backslash circ\; 4\text{'}$.

But a nearer approximation may be obtained, by supposing the second differences of any series to form the arithmetical progression. The sum of the extreme terms $C-2B+A$ and $F-2E+D$ would, therefore, be equal to the sum of the mean terms $D-2C+B$ and $E-2D+C$; whence $F-A=3B+3E-2C-2D$, or $F=A+3(B+E)-2(C+D)$. Thus, the natural sines of the successive arcs 30, 31, 32, 34, and 35 degrees may be easily computed to seven places of figures.

The sines of $31^\backslash circ$ and of $34^\backslash circ$ are here added together, and the sum tripled, and from this amount is taken the double of the sum of the sines of $31^\backslash circ$ and of $33^\backslash circ$; the sine of $30^\backslash circ$ being subtracted from that remainder, leaves finally the sine of $35^\backslash circ$.

Employing the same number of terms of the series, a still closer approximation may be discovered, by considering the third differences only as uniformly progressive. Wherefore the extreme differences $D-3C+3B-A$ and $F-3E+3D-C$ will be together equal to double the middle one, $E-3D+3C-B$, and consequently $F-A=5E-10D+10C-5B$, or $F=A+5(E-B)-10(D-C)$.

Hence the logarithms even of low numbers may be computed exact to eight decimal places. Thus, the logarithms of 150, 151, 152, 153, and 154, being given, that of 155 is found by this process:

The difference between the logarithms of 151 and 154 is here multiplied by 5, and the difference of the logarithms of 152 and 153 is multiplied by 10; and the excess of the former product above the latter being added to the logarithm of 150, gives the logarithm of 155.

It would obviously be preferable, however, to employ the formula in a modified form for interpolation merely. Hence $D=C+\backslash frac\{1\}\{2\}(E-B)+A-F$.

If six terms of the series were given, the seventh could be found to a high degree of accuracy. The sum of the extremes of the progressive third differences being now assumed equal to that of the means, we have $D-3C+3B-A+G-3F+3E-D=E-3D+3C-B+F-3E+3D-C$, and by reduction $G-A=4F-5E-5C-4B$, whence $G=A+4(F-B)-5(E-C)$. It seems unnecessary to subjoin any farther illustrations; but the very simple methods of interpolation now proposed, might be applied with great facility and advantage in various physical researches. In this way, much light may be thrown upon the resistance of fluids, and upon the force, the density, and the component heat of steam, at different temperatures.