OF T H E ROYAL EN G IN EERS
L.L.D. F.K.S. Jtc. &c.
T H IS T A B L E O F L O G A R IT H M S
18 I N S C R I B E D nV
H I 9 F A I T H F U L F R I E N D ,
C H A R L E S B A B B A G E .
P R E F A C E
I n presenting to the public a new table o f Logarithms, two things particularly dem and the attention o f the editor— their correctness, and the facility with which they can be used by com puters. I n order to give confidence in their accuracy, it is necessary th a t the calculations should be true, and th eir results correctly p rin ted ; and th a t the m eans by which this has been attem pted to be accom plished should be fully stated.
T he following arrangem ent was adopted for the purpose o f insur
ing accuracy in the printing.
A copy o f the stereotype logarithm s o f C allet was carefully read over w ith th e folio edition (1794) o f V ega’s logarithm s, to ten figures, and whenever the last figure in C allet’s logarithm s was found to have been increased, it was markedNvith red ink. All th e term inal figures thus m arked were to be printed with a d o t below them.
Types were cast having dots below the figures, and also others with the figures sm aller than those in th e body o f the table T h ese la tte r were to be em ployed whenever th e third figure o f the loga
rithm changed.
T h e proofs o f the p resent tables were read th ree tim e s : 1st, with the m arked copy o f C allet’s lo g a rith m s; 2dly, with a copy o f H u tto n ’s logarithm s, fourth edition, 1804 ; 3dly, w ith a copy o f V ega’s logarithm s, folio, 1794.
T hey were now received from the printer, and were again com pared w ith the logarithm s o f Vega as far as 1 00,000; the last 8,000 being read with those o f C allet öthly, T h e first 20,000 were read with those in the T rigonom etria Artificialis o f Briggs. Folio.
Goudse, 1633.
T hey were n ex t returned to th e printer, and stereotyped, and the proofs from the plates were read ; 6thly, w ith the logarithms o f Vega as far as 17,500 ; 7thly w ith the whole o f the logarithm s o f G ardiner,
vi PR EFA C E.
4to. London, 1742; 8thly, w ith the logarithm s o f Taylor, 4 to . 1792;
and 9thly, by a different set o f readers they were again read with the logarithm s o f Taylor.
I had already experienced the accuracy o f tables printed at the press o f M r. Applegath,* and to th e constant attention of Mr. Cox, the superintendent o f th a t establishm ent, the present work is m aterially indebted.
]n the table o f logarithm s beginning a t 10,000, and term inating a t 108,000, the readers, previous to the stereotyping, detected ten errors o f figures, eight o f which occurred in the first sheet, before th e system o f work, which M r. Cox had arranged, had well got into action. Am ong the differences, there were four errors.
A m ongst the dotted figures, which were m uch m ore difficult to correct, and which the prin ter had fewer m eans o f making accurate, there were found eighteen erroneous.
From the circum stance of these logarithms being wanted for im
mediate use, it was necessary to print them as soon as possible, in order to allow o f tim e for the ink to dry and harden previous to binding. In the readings, after stereotyping, seven errors were discovered in the logarithms, and one in the differences. These have been corrected in the plates. For the whole of the readings subsequent to the third, I am indebted to the kindness of L ieu- tenant-Colonel Colby, under whose direction they were exe
cuted.
In the fourth reading, with Vega’s logarithm s to ten figures, 93 cases were observed in which the last three figures in those tables were 500. As those tables were themselves true to the nearest unit, it was n o t certain w hether, if these logarithm s were given to a g reater num ber o f figures, th e 8 th , 9 th , and 10th m ight not be 499, in which case the last figures o f the table to seven figures, ought n ot to be increased and dotted.
T hus the logarithm s o f I 0117 is
•0050517 500 ;
but the logarithm o f the same num ber to seven figures is not
•0050519, because the logarithm to twelve figures is
•0050517 499 93,
and the eighth figure being really 4, the true logarithm to seven figures is
•0050517.
In seventy-seven instances, the figures succeeding th e seventh were 499 or 500.
In fourteen instances, those figures were 4999 or 5000.
And in two cases, they were 49999 and 50000.
I t becam e necessary therefore, in all these cases, to have the logarithm s carried to m ore th an ten figures.
* N ow that of Mr. Clowe«.
PR EFA C E. v¡¡
In m any o f the earlier instances, the logarithm s were calculated to fifteen figures; b u t as I found it necessary, during the progress o f the impression, to visit Paris, I availed m yself o f th a t opportunity o f consulting the great m anuscript tables preserved a t the Observa
tory, which were calculated under the direction o f M . Prony.
T hrough the kindness o f the M arquis Laplace, President o f the B oard o f Longitude o f France, o f M. Bouvard, and o f o ther members o f th a t distinguished body, I enjoyed every- facility for making the com parisons w hich were requisite for this purpose, as well as for m aking extracts necessary to me for o ther calculations.
A lthough the liberality with which these treasures o f calculation are throw n open to those who m ay wish to avail themselves o f them , and the attentions which such persons receive from every officer of the establishm ent in which they are deposited, deserve to be recom m ended as m odels in all sim ilar cases, I still may be perm itted to express a wish, th a t so vast a m onum ent o f industry and o f science should be rendered m ore useful, as well as m ore perfect and inde
structible, by em bodying it in stereotype plates.
"With regard to those arrangem ents, on which the facility o f using a table depends, I shall offer a few remarks, although a t the hazard o f seeming to bestow more attention than may be necessary on a subject w hich m ay appear o f trifling im portance.
W ith the assistance o f my friend L ieutenant-C olonel Colby, I exam ined a considerable collection o f tables o f different kinds, with the view o f discovering on w hat typographical circum stances perspicuity depended.
W herever we observed any rem arkable for clearness or for obscu
rity, we endeavoured to discover and to note the cause o f such an appearance. From this exam ination resulted th e following collection o f rules, some o f which com m anded im m ediate assent, w hilst others were o f more doubtful propriety.
1st. The clearness or fa c ility o f reading, does not depend on the size o f the type alone, but on the proportion o f the typ e to the in terval between the lines.
2nd. Figures o f the same or nearly the same height, are p re fe r
able to those in tvhich some o f the digits rise above a n d others f a l l below the line, because they interfere less w ith th e space between th e lines.
T h is becomes still m ore necessary, if from any cause it is desir
able to distinguish some o f the figures by m eans o f a bar, a dot, o r any o th e r sign, placed either above or below them .
3rd. The lines d ividing vertical columns should not be placed in the m iddle o f the space between the columns, but should be nearer the
Î
rreceding column. T he effect o f this mode o f placing the vertical ines is to leave a white line a t the com m encem ent o f every column, which n o t only assists in directing th e eye, b u t m akes the adjacentviii PR EFA C E.
dark line m ore distinct. T he same principle is applicable to the case o f a table divided by horizontal lines.
4 th . IV hen some p a rts o f a table are to be separated fr o m the rest more decisively than by the ordinary lines,* a single da rk line is much more conspicuous than two fa in te r lines adjacent to each other; and, i f necessary, f o r fu r th e r distinction, another, a n d much darker line, m ay be employed w ith success.
5th. Those figures which are fir s t sought on entering a table,ought to be so distinguished, either by position or by magnitude, as to strike the eye readily. Since th eir position is not always left to our choice, we m ust sometimes em ploy type o f different kinds or magni
tudes for this purpose. Colour m ight be a convenient distinction in som e cases, b u t the difficulties which occur in printing with two or m ore colours will prevent its frequent em ploym ent. In the present tables, the four first figures o f the num ber sought are printed with large type, in order th a t they may be easily found. F o r the same reason, the figures at the top and bottom o f the page are also larger than the rest. As the natural num bers are in large type, they will readily catch the eye in turning over the pages ; it was therefore thought advisable to p u t a t the (op o f the left-hand page the three first figures o f the first logarithm in th a t page in an equal type, in o rd er to m ake the finding o f the page in which a num ber or a loga
rithm occurs equally easy.
B y this arrangem ent, the right-hand pages will have the num ber nearest to the corner, instead o f the lo g a rith m ; b u t this was not th o u g h t an objection o f sufficient im portance to justify a deviation from the order in which^they are placed on th e opposite pages.
(>th. I n most instances it is better to p r in t the figures denoting the tens, the hundreds, or the thousands, although they m ay remain the same f o r several lines. T he reason o f this rule is, that the eye has not to travel so far to acquire the inform ation sought in the table, f
T he arrangem ent o f the table was a subject o f attentive con
sideration, and 1 availed m yself o f th e observations o f m any o f my friends who had had great experience in the use o f tables. T he m ost difficult question to decide upon, was th e m ode by which the change in the third figure o f a logarithm should be indicated, when it does n o t occur precisely a t the end o f a line.
T h e m ethod used by Callet, o f bringing down the rem ainder o f the line so as to leave a blank space between it and the preceding one, had m any admirers. I t is, however, liable to some objections:
it prevents th e occurrence o f the blank intervals a t regular periods, (usually once in five lines,) an inconvenience o f some m om ent, and
• As in th e case o f the first five colum ns o f a table o f logarithm s, which are separated from the last five.
f T h is rule was not considered until it was too late to comply w ith it in these tables.
P R E FA C E . ix
which increases th e tim e required for using the table. I t is also a t variance with a general principle which I found it necessary to establish, in order to decide on a proper m ode o f m arking this cir
cum stance. T h e principle alluded to is, th a t
7th. W hatever mode be adopted f o r m arking the change o f the th ird figure, it ought to be o f sveh a nature th a t i f the fo u r last figures o f any logarithm be selected, in the m iddle or in any p a r t o f a page, it shall be immediately visible w ithout reference to any other p a rt o f the table, whether the th ird fig u re has changed or not.
T he blank space m arking th e break in the line, em ployed by Callet, agrees with this principle a t the com m encem ent of his tables; b u t throughout th e g reater part, there are blank spaces which do not indicate any change o f the third figure. A nother objection to his mode is, th a t th e figures o f the logarithm s cease to be opposite those o f their corresponding natural num bers.
T h e m ethod em ployed by D r. H u tto n consists in placing a bar above the first figure after th e change takes p la c e : this is inad
missible, because it gives no notice o f the change in the succeeding lo g arith m s: the sam e objection applies to those tables in which a star is prefixed to the logarithm next to th a t in w hich the change occurs. In the logarithm ic tables o f Vega the star is used, b u t it is continued before th e fourth figure o f each succeeding logarithm to the end o f the line. T here is no objection to this in point o f principle, and the only inconvenience which attends it, is th a t it increases the breadth o f th e page, a circum stance which could n o t be adm itted in the present tables, w ithout sacrificing the colum n for converting sexagesimal arcs into seconds.
T h e utility o f this la tte r p art o f the table was so sensibly felt, th a t the plan o f em ploying a figure o f about h a lf th e usual size was preferred, and by continuing the same type to the end o f the line in which the change in the third figure had occurred, the prin
ciple w hich has been laid down was adhered to, and sufficient space was preserved for the colum n alluded to. I t will be observed, th a t these sm all figures do not stand precisely in the m iddle o f the space allotted to th e m ; there are two reasons for this deviation: the proxim ity o f a small figure to a large one m akes the difference of the two types m ore conspicuous, and the w hite vertical line preceding each logarithm is rendered more apparent by the space which thus occurs on the left side o f every figure after the change has taken place.
By the adoption o f this small figure, another advantage has been g a in e d ; the pages are now divided a t every fifth line by a blank space, which enables the reader m ore easily to catch any logarithm he may be seeking. I t is usual in m any tables to m ark these in
tervals by rules instead o f spaces ; neither o f these m ethods appear to possess any very decided advantages over the other. T hey may, however, be em ployed in conjunction, and the result is a greater
X PR EFA C E.
degree o f distinctness than arises from the use o f either separately.
I f a rule is placed a t intervals o f ten lines, and if these tens are subdivided by blank spaces into periods o f five lines, th e form o f the table will be considerably improved. T h e reason which has induced me not to adopt this im provem ent in the present tables, is to be found in the difficulty o f its typographical execution. W hen two rules, a vertical and an horizontal one, intersect each other, one o f them m ust be cu t so as to adm it o f the passage o f the o th e r;
and it will be observed, th a t whenever this occurs the two printed lines never appear to cross each other, b u t a small break is left at th e intersection ; this m ay be noticed either a t the top or bottom o f the page, in the continuation o f the lines which separate the five first colum ns from the five last o f the logarithm s. In con
sequence o f this difficulty the rules between every ten th line have been om itted, and th e space left is sufficient to adm it o f a line being drawn with ink at those intervals, a plan which I have adopted in my own copy, and which I should recom m end to all who have frequent occasion for the use o f these tables.
8th. ff'henever additional inform ation can be communicated in a table without increasing its bulk, or adding much to its expense, it ought alw ays to be given; unless it is o f such a nature as to distract the attention too much fr o m the p a r t most freq u en tly used. In com pliance with this principle, it was my wish to m ark those term inal figures o f the logarithm s which had been m ade true to the nearest u n it by some peculiarity which, whilst it should n o t render them conspicuous, should yet distinguish them sufficiently w henever such accuracy should be desirable. A sim ple d o t attached to a le tte r ’ affords the sm allest visible appendage, and I had at first proposed to place a d o t over each letter, which had been increased by a u n it:
this position was, however, given up on further consideration, be
cause th e same sign has in some cases been em ployed to denote recurring decimals. Its position was therefore changed, and those figures which have been increased are distinguished by a dot im m e
diately below them .
A t the left side o f each page o f the logarithm ic table, and sepa
rated from it by a broad dark line, are placed the degrees, m inutes, and seconds corresponding to the natural num bers adjacent. T he utility o f this addition is so great, and its em ploym ent so frequent, th a t it was thought desirable to add it even a t a considerable increase o f expense. T he circum stance o f this table being arranged in pages o f fifty lines each, a plan which was required by o th er reasons, has caused the whole degrees and m inutes not to term inate a t the bottom o f each page. T his inconvenience was considered o f less im portance than those w hich would have arisen from em ploying a page o f sixty lines. If, however, any error should be feared from overlooking the change o f th e m inutes in the first colum n, or of the degrees in the second, which m ay occur in the
PR EFA C E. xi middle o f a page, it may be pointed o u t by drawing a line in red ink under the figure where the change takes place in the first colum n, and one in black ink a t sim ilar places in the second.
In order to prevent the inconvenience o f the tables o f arcs catching the eye o f persons only using the logarithm s, it has been iut in different and m uch sm aller type, and has been com pletely eparated from the page by a broad dark line.
9th. The different tables in a volume ought to be distinguished fr o m each other by the art o f the printer, in such a m anner that a ny one m ay, fr o m its peculiarity, be readily distinguished in turning the pages over rapidly. This m ay be accom plished— by enclosing some o f them writh rules— by putting broad borders o f various kinds at the tops and sides o f others— or by having them printed in a dif
ferent coloured ink.
Every table ought to have a running title at the top expressed as concisely as possible, and when the formula by w hich it is con
structed is simple, it would be advantageous to give it on each page.
T here are some causes o f indistinctness in tables which arise from o ther sources. I t frequently happens, th at
10th. T he impression o f the figures on one page is reversed on the opposite: this sometimes arises from the volume having been bound before th e ink was quite dry, a fault which should be m ost carefully avoided. I t is, however, more frequently caused by the bad quality of the ink em ployed by the printer.
A nother defect arises when
11th. The transparency o f the p a p er adm its the figures on the reverse side to appear th rough.
In order to try w hether
12th. Coloured paper is more favourable to distinctness than white, I had a page set up, and printed on paper o f various colours and sh a d e s: alm ost all those whom 1 consulted agreed with me in giving the preference to the coloured papers, b u t the particular tin t was not so unanim ously fixed upon. Yellow appeared to have the preference, and it is th a t which I have chosen for the first im pres
sion. T h e tint is a t first considerably too deep, b u t it fades on ex
posure to the lig h t: I should therefore recom m end, th a t previously to binding this volume the sheets should be exposed for several days to the action o f the sun
I t is not probable th a t the colour first selected should be found the b e s t; we m ust wait the result o f time and experience to d eter
I t is not probable th a t the colour first selected should be found the b e s t; we m ust wait the result o f time and experience to d eter