# 1911 Encyclopædia Britannica/Logarithm

**LOGARITHM** (from Gr. λόγος, word, ratio, and ἀριθμός, number), in mathematics, a word invented by John Napier to denote a particular class of function discovered by him, and which may be defined as follows: if *a*, *x*, *m* are any three quantities satisfying the equation a^{x} = *m*, then *a* is called the base, and *x* is said to be the logarithm of *m* to the base *a*. This relation between *x*, *a*, *m*, may be expressed also by the equation *x* = log_{a} *m*.

*Properties.*—The principal properties of logarithms are given by the equations

log_{a} (mn) = log_{a} m + log_{a} n, |
log_{a} (m/n) = log_{a} m − log_{a} n, |

log_{a} m^{r} = r log_{a} m, |
log_{a} ^{r}√ m = (1/r) log_{a} m, |

which may be readily deduced from the definition of a logarithm.
It follows from these equations that the logarithm of the product
of any number of quantities is equal to the sum of the logarithms
of the quantities, that the logarithm of the quotient of two
quantities is equal to the logarithm of the numerator diminished
by the logarithm of the denominator, that the logarithm of the
*r* th power of a quantity is equal to *r* times the logarithm of the
quantity, and that the logarithm of the *r*th root of a quantity
is equal to (1/*r*)th of the logarithm of the quantity.

Logarithms were originally invented for the sake of abbreviating
arithmetical calculations, as by their means the operations
of multiplication and division may be replaced by those of
addition and subtraction, and the operations of raising to powers
and extraction of roots by those of multiplication and division.
For the purpose of thus simplifying the operations of arithmetic,
the base is taken to be 10, and use is made of tables of
logarithms in which the values of *x*, the logarithm, corresponding
to values of *m*, the number, are tabulated. The
logarithm is also a function of frequent occurrence in analysis,
being regarded as a known and recognized function like sin *x* or
tan *x*; but in mathematical investigations the base generally
employed is not 10, but a certain quantity usually denoted by the
letter *e*, of value 2.71828 18284....

Thus in arithmetical calculations if the base is not expressed
it is understood to be 10, so that log *m* denotes log_{10} *m*; but in
analytical formulae it is understood to be *e*.

The logarithms to base 10 of the first twelve numbers to 7 places of decimals are

log 1 = 0.0000000 | log 5 = 0.6989700 | log 9 = 0.9542425 |

log 2 = 0.3010300 | log 6 = 0.7781513 | log 10 = 1.0000000 |

log 3 = 0.4771213 | log 7 = 0.8450980 | log 11 = 1.0413927 |

log 4 = 0.6020600 | log 8 = 0.9030900 | log 12 = 1.0791812 |

The meaning of these results is that

1 = 100, | 2 = 100.3010300, | 3 = 100.4771213, ... |

10 = 10^{1}, | 11 = 101.0413927, | 12 = 101.0791812. |

The integral part of a logarithm is called the index or characteristic, and the fractional part the mantissa. When the base is 10, the logarithms of all numbers in which the digits are the same, no matter where the decimal point may be, have the same mantissa; thus, for example,

In the case of fractional numbers (*i.e.* numbers in which the
integral part is 0) the mantissa is still kept positive, so that,
for example,

the minus sign being usually written over the characteristic, and not before it, to indicate that the characteristic only, and not the whole expression, is negative; thus

The fact that when the base is 10 the mantissa of the logarithm
is independent of the position of the decimal point in the number
affords the chief reason for the choice of 10 as base. The explanation
of this property of the base 10 is evident, for a change
in the position of the decimal points amounts to multiplication
or division by some power of 10, and this corresponds to the
addition or subtraction of some integer in the case of the
logarithm, the mantissa therefore remaining intact. It should
be mentioned that in most tables of trigonometrical functions,
the number 10 is added to all the logarithms in the table in order
to avoid the use of negative characteristics, so that the characteristic
9 denotes in reality 1, 8 denotes 2, 10 denotes 0, &c.
Logarithms thus increased are frequently referred to for the sake
of distinction as *tabular logarithms*, so that the tabular logarithm
= the true logarithm + 10.

In tables of logarithms of numbers to base 10 the mantissa only is in general tabulated, as the characteristic of the logarithm of a number can always be written down at sight, the rule being that, if the number is greater than unity, the characteristic is less by unity than the number of digits in the integral portion of it, and that if the number is less than unity the characteristic is negative, and is greater by unity than the number of ciphers between the decimal point and the first significant figure.

It follows very simply from the definition of a logarithm that

_{a}

*b*× logb

*a*= 1, logb

*m*= log

_{a}

*m*× (1/log

_{a}

*b*).

The second of these relations is an important one, as it shows
that from a table of logarithms to base *a*, the corresponding
table of logarithms to base *b* may be deduced by multiplying all
the logarithms in the former by the constant multiplier 1/log_{a} *b*,
which is called the *modulus* of the system whose base is *b* with
respect to the system whose base is *a*.

The two systems of logarithms for which extensive tables
have been calculated are the Napierian, or hyperbolic, or natural
system, of which the base is e, and the Briggian, or decimal, or
common system, of which the base is 10; and we see that the
logarithms in the latter system may be deduced from those in the
former by multiplication by the constant multiplier 1/log_{e} 10,
which is called the modulus of the common system of logarithms.
The numerical value of this modulus is 0.43429 44819 03251
82765 11289 ..., and the value of its reciprocal, log^{e} 10 (by
multiplication by which Briggian logarithms may be converted
into Napierian logarithms) is 2.30258 50929 94045 68401
79914 ....

The quantity denoted by *e* is the series,

1 + | 1 | + | 1 | + | 1 | + | 1 | + ... |

1 | 1·2 | 1·2·3 | 1·2·3·4 |

the numerical value of which is,

*The logarithmic Function.*—The mathematical function log *x* or
log_{e} *x* is one of the small group of transcendental functions, consisting
only of the circular functions (direct and inverse) sin *x*, cos *x*,
&c., arc sin *x* or sin^{−1} *x*,&c., log *x* and *e*^{x} which are universally treated
in analysis as known functions. The notation log *x* is generally
employed in English and American works, but on the continent of
Europe writers usually denote the function by *lx* or lg *x*. The
logarithmic function is most naturally introduced into analysis by
the equation

. |

This equation defines log *x* for positive values of *x*; if *x* ≤ 0 the
formula ceases to have any meaning. Thus log *x* is the integral
function of 1/*x*, and it can be shown that log *x* is a genuinely new
transcendent, not expressible in finite terms by means of functions
such as algebraical or circular functions. A connexion with the
circular functions, however, appears later when the definition of
log *x* is extended to complex values of *x*.

A relation which is of historical interest connects the logarithmic
function with the quadrature of the hyperbola, for, by considering
the equation of the hyperbola in the form xy = const., it is evident
that the area included between the arc of a hyperbola, its nearest
asymptote, and two ordinates drawn parallel to the other asymptote
from points on the first asymptote distant *a* and *b* from their point
of intersection, is proportional to log *b*/*a*.

The following fundamental properties of log *x* are readily deducible
from the definition

(i.) log *xy* = log *x* + log *y*.

(ii.) Limit of (x^{h} − 1)/*h* = log *x*, when *h* is indefinitely diminished.

Either of these properties might be taken as itself the definition of
log *x*.

There is no series for log *x* proceeding either by ascending or
descending powers of *x*, but there is an expansion for log (1 + *x*), viz.

*x*) =

*x*− 12

*x*

^{2}+ 13

*x*

^{3}− 14

*x*

^{4}+ ...;

the series, however, is convergent for real values of *x* only when *x* lies
between +1 and −1. Other formulae which are deducible from this
equation are given in the portion of this article relating to the calculation
of logarithms.

The function log *x* as *x* increases from 0 towards ∞ steadily increases
from −∞ towards +∞. It has the important property that
it tends to infinity with *x*, but more slowly than any power of *x*, *i.e.*
that *x*−m log *x* tends to zero as *x* tends to ∞ for every positive value
of *m* however small.

The *exponential function*, exp *x*, may be defined as the inverse of
the logarithm: thus *x* = exp *y* if *y* = log *x*. It is positive for all values
of *y* and increases steadily from 0 toward ∞ as *y* increases from -∞
towards +∞. As *y* tends towards ∞, exp *y* tends towards ∞
more rapidly than any power of *y*.

The exponential function possesses the properties

(i.) | exp (x + y) = exp x × exp y. |

(ii.) | (d/dx) exp x = exp x. |

(iii.) | exp x = 1 + x + x^{2}/2! + x^{3}/3! + ... |

From (i.) and (ii.) it may be deduced that

*x*= (1 + 1 + 1/2! + 1/3! + ... )

^{x},

where the right-hand side denotes the positive xth power of the
number 1 + 1 + 1/2! + 1/3! + ... usually denoted by *e*. It is customary,
therefore, to denote the exponential function by *e*^{x} and the
result

^{x}= 1 +

*x*+

*x*

^{2}/2! +

*x*

^{3}/3! ...

is known as the *exponential theorem*.

The definitions of the logarithmic and exponential functions may
be extended to complex values of *x*. Thus if *x* = ξ + iη

where the path of integration in the plane of the complex variable t
is any curve which does not pass through the origin; but now log x
is not a uniform function, that is to say, if *x* describes a closed curve
it does not follow that log *x* also describes a closed curve: in fact
we have

^{2}+ η

^{2}) + i(α + 2nπ),

where α is the numerically least angle whose cosine and sine are
ξ/√(ξ^{2} + η^{2}) and η/√(ξ^{2} + η^{2}), and n denotes any integer. Thus even
when the argument is real log *x* has an infinite number of values; for
putting η = 0 and taking ξ positive, in which case α = 0, we obtain for
log ξ the infinite system of values log ξ + 2nπi. It follows from this
property of the function that we cannot have for log *x* a series which
shall be convergent for all values of *x*, as is the case with sin *x* and
cos *x*, for such a series could only represent a uniform function, and in
fact the equation

*x*) =

*x*− 12

*x*

^{2}+ 13

*x*

^{3}− 14

*x*

^{4}+ ...

is true only when the analytical modulus of *x* is less than unity.
The exponential function, which may still be defined as the inverse
of the logarithmic function, is, on the other hand, a uniform function
of *x*, and its fundamental properties may be stated in the same form
as for real values of *x*. Also

*e*ξ (cos η +

*i*sin η).

An alternative method of developing the theory of the exponential function is to start from the definition

*x*= 1 +

*x*+

*x*

^{2}/2! +

*x*

^{3}/3! + ...,

the series on the right-hand being convergent for all values of *x* and
therefore defining an analytical function of *x* which is uniform and
regular all over the plane.

*Invention and Early History of Logarithms.*—The invention of
logarithms has been accorded to John Napier, baron of Merchiston
in Scotland, with a unanimity which is rare with regard to
important scientific discoveries: in fact, with the exception of
the tables of Justus Byrgius, which will be referred to further on,
there seems to have been no other mathematician of the time
whose mind had conceived the principle on which logarithms
depend, and no partial anticipations of the discovery are met
with in previous writers.

The first announcement of the invention was made in Napier’s
*Mirifici Logarithmorum Canonis Descriptio ...* (Edinburgh,
1614). The work is a small quarto containing fifty-seven pages
of explanatory matter and a table of ninety pages (see Napier, John).
The nature of logarithms is explained by reference to
the motion of points in a straight line, and the principle upon
which they are based is that of the correspondence of a geometrical
and an arithmetical series of numbers. The table gives
the logarithms of sines for every minute of seven figures; it is
arranged semi-quadrantally, so that the *differentiae*, which are
the differences of the two logarithms in the same line, are the
logarithms of the tangents. Napier’s logarithms are not the
logarithms now termed Napierian or hyperbolic, that is to say,
logarithms to the base *e* where *e* = 2.7182818...; the relation
between N (a sine) and L its logarithm, as defined in the *Canonis*
*Descriptio*, being N = 10^{7} *e*−L/(10^{7}), so that (ignoring the factors 10^{7},
the effect of which is to render sines and logarithms integral to
7 figures), the base is *e*^{−1}. Napier’s logarithms decrease as the
sines increase. If l denotes the logarithm to base *e* (that is, the
so-called “Napierian” or hyperbolic logarithm) and L denotes,
as above, “Napier’s” logarithm, the connexion between *l* and
L is expressed by

^{7}log

_{e}10

^{7}− 10

^{7}

*l*or

*e*

^{l}= 10

^{7}

*e*−L/(10

^{7})

Napier’s work (which will henceforth in this article be referred
to as the *Descriptio*) immediately on its appearance in 1614
attracted the attention of perhaps the two most eminent English
mathematicians then living—Edward Wright and Henry Briggs.
The former translated the work into English; the latter was
concerned with Napier in the change of the logarithms from those
originally invented to decimal or common logarithms, and it is
to him that the original calculation of the logarithmic tables now
in use is mainly due. Both Napier and Wright died soon after
the publication of the *Descriptio*, the date of Wright’s death
being 1615 and that of Napier 1617, but Briggs lived until 1631.
Edward Wright, who was a fellow of Caius College, Cambridge,
occupies a conspicuous place in the history of navigation. In
1599 he published *Certaine errors in Navigation detected and*
*corrected*, and he was the author of other works; to him also is
chiefly due the invention of the method known as Mercator’s
sailing. He at once saw the value of logarithms as an aid to
navigation, and lost no time in preparing a translation, which
he submitted to Napier himself. The preface to Wright’s
edition consists of a translation of the preface to the *Descriptio*,
together with the addition of the following sentences written by
Napier himself: “But now some of our countreymen in this
Island well affected to these studies, and the more publique
good, procured a most learned Mathematician to translate the
same into our vulgar English tongue, who after he had finished it,
sent the Coppy of it to me, to bee seene and considered on by
myselfe. I having most willingly and gladly done the same, finde
it to bee most exact and precisely conformable to my minde and
the originall. Therefore it may please you who are inclined to
these studies, to receive it from me and the Translator, with
as much good will as we recommend it unto you.” There is a
short “preface to the reader” by Briggs, and a description of a
triangular diagram invented by Wright for finding the proportional
parts. The table is printed to one figure less than in the
*Descriptio*. Edward Wright died, as has been mentioned, in
1615, and his son, Samuel Wright, in the preface states that his
father “gave much commendation of this work (and often in my
hearing) as of very great use to mariners”; and with respect to
the translation he says that “shortly after he had it returned
out of Scotland, it pleased God to call him away afore he could
publish it.” The translation was published in 1616. It was also
reissued with a new title-page in 1618.

Henry Briggs, then professor of geometry at Gresham College,
London, and afterwards Savilian professor of geometry at Oxford,
welcomed the *Descriptio* with enthusiasm. In a letter to Archbishop
Usher, dated Gresham House, March 10, 1615, he wrote,
“Napper, lord of Markinston, hath set my head and hands a
work with his new and admirable logarithms. I hope to see him
this summer, if it please God, for I never saw book which pleased
me better, or made me more wonder.^{[1]} I purpose to discourse
with him concerning eclipses, for what is there which we may not
hope for at his hands,” and he also states “that he was wholly
taken up and employed about the noble invention of logarithms
lately discovered.” Briggs accordingly visited Napier in 1615,
and stayed with him a whole month.^{[2]} He brought with him some
calculations he had made, and suggested to Napier the advantages
that would result from the choice of 10 as a base, an improvement
which he had explained in his lectures at Gresham College, and
on which he had written to Napier. Napier said that he had
already thought of the change, and pointed out a further improvement,
viz., that the characteristics of numbers greater
than unity should be positive and not negative, as suggested by
Briggs. In 1616 Briggs again visited Napier and showed him the
work he had accomplished, and, he says, he would gladly have
paid him a third visit in 1617 had Napier’s life been spared.

Briggs’s *Logarithmorum chilias prima*, which contains the first
published table of decimal or common logarithms, is only a
small octavo tract of sixteen pages, and gives the logarithms
of numbers from unity to 1000 to 14 places of decimals. It was
published, probably privately, in 1617, after Napier’s death,^{[3]} and
there is no author’s name, place or date. The date of publication
is, however, fixed as 1617 by a letter from Sir Henry Bourchier
to Usher, dated December 6, 1617, containing the passage—“Our
kind friend, Mr Briggs, hath lately published a supplement
to the most excellent tables of logarithms, which I presume he
has sent to you.” Briggs’s tract of 1617 is extremely rare, and
has generally been ignored or incorrectly described. Hutton
erroneously states that it contains the logarithms to 8 places,
and his account has been followed by most writers. There is a
copy in the British Museum.

Briggs continued to labour assiduously at the calculation of
logarithms, and in 1624 published his *Arithmetica logarithmica*,
a folio work containing the logarithms of the numbers from l
to 20,000, and from 90,000 to 100,000 (and in some copies to
101,000) to 14 places of decimals. The table occupies 300 pages,
and there is an introduction of 88 pages relating to the mode of
calculation, and the applications of logarithms.

There was thus left a gap between 20,000 and 90,000, which
was filled up by Adrian Vlacq (or Ulaccus), who published at
Gouda, in Holland, in 1628, a table containing the logarithms
of the numbers from unity to 100,000 to 10 places of decimals.
Having calculated 70,000 logarithms and copied only 30,000,
Vlacq would have been quite entitled to have called his a new
work. He designates it, however, only a second edition of
Briggs’s *Arithmetica logarithmica*, the title running *Arithmetica*
*logarithmica sive Logarithmorum Chiliades centum, ... editio*
*secunda aucta per Adrianum Vlacq, Goudanum*. This table of
Vlacq’s was published, with an English explanation prefixed,
at London in 1631 under the title *Logarithmicall Arithmetike ...*
*London, printed by George Miller*, 1631. There are also copies
with the title-page and introduction in French and in Dutch
(Gouda, 1628).

Briggs had himself been engaged in filling up the gap, and in a letter to John Pell, written after the publication of Vlacq’s work, and dated October 25, 1628, he says:—

“My desire was to have those chiliades that are wantinge betwixt 20 and 90 calculated and printed, and I had done them all almost by my selfe, and by some frendes whom my rules had sufficiently informed, and by agreement the busines was conveniently parted amongst us; but I am eased of that charge and care by one Adrian Vlacque, an Hollander, who hathe done all the whole hundred chiliades and printed them in Latin, Dutche and Frenche, 1000 bookes in these 3 languages, and hathe sould them almost all. But he hathe cutt off 4 of my figures throughout; and hathe left out my dedication, and to the reader, and two chapters the 12 and 13, in the rest he hath not varied from me at all.”

The original calculation of the logarithms of numbers from unity to 101,000 was thus performed by Briggs and Vlacq between 1615 and 1628. Vlacq’s table is that from which all the hundreds of tables of logarithms that have subsequently appeared have been derived. It contains of course many errors, which were gradually discovered and corrected in the course of the next two hundred and fifty years.

The first calculation or publication of Briggian or common
logarithms of trigonometrical functions was made in 1620 by
Edmund Gunter, who was Briggs’s colleague as professor of
astronomy in Gresham College. The title of Gunter’s book,
which is very scarce, is *Canon triangulorum*, and it contains
logarithmic sines and tangents for every minute of the quadrant
to 7 places of decimals.

The next publication was due to Vlacq, who appended to his
logarithms of numbers in the *Arithmetica logarithmica* of 1628
a table giving log sines, tangents and secants for every minute
of the quadrant to 10 places; these were obtained by calculating
the logarithms of the natural sines, &c. given in the *Thesaurus*
*mathematicus* of Pitiscus (1613).

During the last years of his life Briggs devoted himself to the
calculation of logarithmic sines, &c. and at the time of his death
in 1631 he had all but completed a logarithmic canon to every
hundredth of a degree. This work was published by Vlacq at
his own expense at Gouda in 1633, under the title *Trigonometria*
*Britannica*. It contains log sines (to 14 places) and tangents (to
10 places), besides natural sines, tangents and secants, at intervals
of a hundredth of a degree. In the same year Vlacq published
at Gouda his *Trigonometria artificialis*, giving log sines and
tangents to every 10 seconds of the quadrant to 10 places.
This work also contains the logarithms of numbers from unity
to 20,000 taken from the *Arithmetica logarithmica* of 1628.
Briggs appreciated clearly the advantages of a centesimal division
of the quadrant, and by dividing the degree into hundredth parts
instead of into minutes, made a step towards a reformation in
this respect, and but for the appearance of Vlacq’s work the
decimal division of the degree might have become recognized,
as is now the case with the corresponding division of the second.
The calculation of the logarithms not only of numbers but also
of the trigonometrical functions is therefore due to Briggs and
Vlacq; and the results contained in their four fundamental
works—*Arithmetica logarithmica* (Briggs), 1624; *Arithmetica*
*logarithmica* (Vlacq), 1628; *Trigonometria Britannica* (Briggs),
1633; *Trigonometria artificialis* (Vlacq), 1633—have not been
superseded by any subsequent calculations.

In the preceding paragraphs an account has been given of the actual announcement of the invention of logarithms and of the calculation of the tables. It now remains to refer in more detail to the invention itself and to examine the claims of Napier and Briggs to the capital improvement involved in the change from Napier’s original logarithms to logarithms to the base 10.

The *Descriptio* contained only an explanation of the use of
the logarithms without any account of the manner in which
the canon was constructed. In an “Admonitio” on the seventh
page Napier states that, although in that place the mode of construction
should be explained, he proceeds at once to the use
of the logarithms, “ut praelibatis prius usu, et rei utilitate,
caetera aut magis placeant posthac edenda, aut minus saltem
displiceant silentio sepulta.” He awaits therefore the judgment
and censure of the learned “priusquam caetera in lucem temerè
prolata lividorum detrectationi exponantur”; and in an
“Admonitio” on the last page of the book he states that he
will publish the mode of construction of the canon “si huius
inventi usum eruditis gratum fore intellexero.” Napier, however,
did not live to keep this promise. In 1617 he published a small
work entitled *Rabdologia* relating to mechanical methods of
performing multiplications and divisions, and in the same year
he died.

The proposed work was published in 1619 by Robert Napier,
his second son by his second marriage, under the title *Mirifici*
*logarithmorum canonis constructio*.... It consists of two
pages of preface followed by sixty-seven pages of text. In the
preface Robert Napier says that he has been assured from undoubted
authority that the new invention is much thought of
by the ablest mathematicians, and that nothing would delight
them more than the publication of the mode of construction
of the canon. He therefore issues the work to satisfy their
desires, although, he states, it is manifest that it would have
seen the light in a far more perfect state if his father could
have put the finishing touches to it; and he mentions that,
in the opinion of the best judges, his father possessed, among
other most excellent gifts, in the highest degree the power of
explaining the most difficult matters by a certain and easy method
in the fewest possible words.

It is important to notice that in the *Constructio* logarithms
are called artificial numbers; and Robert Napier states that the
work was composed several years (*aliquot annos*) before Napier
had invented the name logarithm. The *Constructio* therefore
may have been written a good many years previous to the
publication of the *Descriptio* in 1614.

Passing now to the invention of common or decimal logarithms,
that is, to the transition from the logarithms originally invented
by Napier to logarithms to the base 10, the first allusion to a
change of system occurs in the “Admonitio” on the last page
of the *Descriptio* (1614), the concluding paragraph of which is
“Verùm si huius inventi usum eruditis gratum fore intellexero,
dabo fortasse brevi (Deo aspirante) rationem ac methodum aut
hunc canonem emendandi, aut emendatiorem de novo condendi,
ut ita plurium Logistarum diligentia, limatior tandem et accuratior,
quàm unius opera fieri potuit, in lucem prodeat. Nihil in ortu
perfectum.” In some copies, however, this “Admonitio” is
absent. In Wright’s translation of 1616 Napier has added the
sentence—“But because the addition and subtraction of these
former numbers may seeme somewhat painfull, I intend (if it
shall please God) in a second Edition, to set out such Logarithmes
as shall make those numbers above written to fall upon decimal
numbers, such as 100,000,000, 200,000,000, 300,000,000, &c.,
which are easie to be added or abated to or from any other
number” (p. 19); and in the dedication of the *Rabdologia* (1617)
he wrote “Quorum quidem Logarithmorum speciem aliam multò
praestantiorem nunc etiam invenimus, & creandi methodum,
unà cum eorum usu (si Deus longiorem vitae & valetudinis
usuram concesserit) evulgare statuimus; ipsam autem novi
canonis supputationem, ob infirmam corporis nostri valetudinem,
viris in hoc studii genere versatis relinquimus: imprimis verò
doctissimo viro D. Henrico Briggio Londini publico Geometriae
Professori, et amico mihi longè charissimo.”

Briggs in the short preface to his *Logarithmorum chilias*
(1617) states that the reason why his logarithms are different
from those introduced by Napier “sperandum, ejus librum
posthumum, abunde nobis propediem satisfacturum.” The
“liber posthumus” was the *Constructio* (1619), in the preface
to which Robert Napier states that he has added an appendix
relating to another and more excellent species of logarithms, referred
to by the inventor himself in the *Rabdologia*, and in which
the logarithm of unity is 0. He also mentions that he has
published some remarks upon the propositions in spherical
trigonometry and upon the new species of logarithms by Henry
Briggs, “qui novi hujus Canonis supputandi laborem gravissimum,
pro singulari amicitiâ quae illi cum Patre meo L. M. intercessit,
animo libentissimo in se suscepit; creandi methodo, et usuum
explanatione Inventori relictis. Nunc autem ipso ex hâc vitâ
evocato, totius negotii onus doctissimi Briggii humeris incumbere,
et Sparta haec ornanda illi sorte quadam obtigisse videtur.”

In the address prefixed to the *Arithmetica logarithmica* (1625)
Briggs bids the reader not to be surprised that these logarithms
are different from those published in the *Descriptio*:—

“Ego enim, cum meis auditoribus Londini, publice in Collegio Greshamensi horum doctrinam explicarem; animadverti multo futurum commodius, si Logarithmus sinus totius servaretur 0 (ut in Canone mirifico), Logarithmus autem partis decimae ejusdem sinus totius, nempe sinus 5 graduum, 44, m. 21, s., esset 10000000000. atque ea de re scripsi statim ad ipsum authorem, et quamprimum per anni tempus, et vacationem a publico docendi munere licuit, profectus sum Edinburgum; ubi humanissime ab eo acceptus haesi per integrum mensem. Cum autem inter nos de horum mutatione sermo haberetur; ille se idem dudum sensisse, et cupivisse dicebat: veruntamen istos, quos jam paraverat edendos curasse, donec alios, si per negotia et valetudinem liceret, magis commodos confecisset. Istam autem mutationem ita faciendam censebat, ut 0 esset Logarithmus unitatis, et 10000000000 sinus totius: quod ego longe commodissimum esse non potui non agnoscere. Coepi igitur, ejus hortatu, rejectis illis quos anteà paraveram, de horum calculo serio cogitare; et sequenti aestate iterum profectus Edinburgum, horum quos hic exhibeo praecipuos, illi ostendi, idem etiam tertia aestate libentissime facturus, si Deus illum nobis tamdiu superstitem esse voluisset.”

There is also a reference to the change of the logarithms on the title-page of the work.

These extracts contain all the original statements made by
Napier, Robert Napier and Briggs which have reference to the
origin of decimal logarithms. It will be seen that they are all
in perfect agreement. Briggs pointed out in his lectures at
Gresham College that it would be more convenient that 0 should
stand for the logarithm of the whole sine as in the *Descriptio*,
but that the logarithm of the tenth part of the whole sine should
be 10,000,000,000. He wrote also to Napier at once; and as
soon as he could he went to Edinburgh to visit him, where, as
he was most hospitably received by him, he remained for a
whole month. When they conversed about the change of system,
Napier said that he had perceived and desired the same thing,
but that he had published the tables which he had already prepared,
so that they might be used until he could construct others
more convenient. But he considered that the change ought
to be so made that 0 should be the logarithm of unity and
10,000,000,000 that of the whole sine, which Briggs could not
but admit was by far the most convenient of all. Rejecting
therefore, those which he had prepared already, Briggs began,
at Napier’s advice, to consider seriously the question of the
calculation of new tables. In the following summer he went
to Edinburgh and showed Napier the principal portion of the
logarithms which he published in 1624. These probably included
the logarithms of the first chiliad which he published in 1617.

It has been thought necessary to give in detail the facts relating
to the conversion of the logarithms, as unfortunately Charles
Hutton in his history of logarithms, which was prefixed to the
early editions of his *Mathematical Tables*, and was also published
as one of his *Mathematical Tracts*, has charged Napier with want
of candour in not telling the world of Briggs’s share in the change
of system, and he expresses the suspicion that “Napier was
desirous that the world should ascribe to him alone the merit
of this very useful improvement of the logarithms.” According
to Hutton’s view, the words, “*it is to be hoped* that his posthumous
work” ... which occur in the preface to the *Chilias*, were a
modest hint that the share Briggs had had in changing the
logarithms should be mentioned, and that, as no attention was
paid to it, he himself gave the account which appears in the
*Arithmetica* of 1624. There seems, however, no ground whatever
for supposing that Briggs meant to express anything beyond his
hope that the reason for the alteration would be explained in
the posthumous work; and in his own account, written seven
years after Napier’s death and five years after the appearance
of the work itself, he shows no injured feeling whatever, but
even goes out of his way to explain that he abandoned his own
proposed alteration in favour of Napier’s, and, rejecting the
tables he had already constructed, began to consider the calculation
of new ones. The facts, as stated by Napier and Briggs,
are in complete accordance, and the friendship existing between
them was perfect and unbroken to the last. Briggs assisted
Robert Napier in the editing of the “posthumous work,” the
*Constructio*, and in the account he gives of the alteration of the
logarithms in the *Arithmetica* of 1624 he seems to have been
more anxious that justice should be done to Napier than to himself;
while on the other hand Napier received Briggs most
hospitably and refers to him as “amico mihi longè charissimo.”

Hutton’s suggestions are all the more to be regretted as they
occur as a history which is the result of a good deal of investigation
and which for years was referred to as an authority by many
writers. His prejudice against Napier naturally produced
retaliation, and Mark Napier in defending his ancestor has fallen
into the opposite extreme of attempting to reduce Briggs to
the level of a mere computer. In connexion with this controversy
it should be noticed that the “Admonitio” on the last page
of the *Descriptio*, containing the reference to the new logarithms,
does not occur in all the copies. It is printed on the back of
the last page of the table itself, and so cannot have been torn
out from the copies that are without it. As there could have
been no reason for omitting it after it had once appeared, we
may assume that the copies which do not have it are those which
were first issued. It is probable, therefore, that Briggs’s copy
contained no reference to the change, and it is even possible
that the “Admonitio” may have been added after Briggs had
communicated with Napier. As special attention has not been
drawn to the fact that some copies have the “Admonitio”
and some have not, different writers have assumed that Briggs
did or did not know of the promise contained in the “Admonitio”
according as it was present or absent in the copies they had
themselves referred to, and this has given rise to some confusion.
It may also be remarked that the date frequently assigned to
Briggs’s first visit to Napier is 1616, and not 1615 as stated above,
the reason being that Napier was generally supposed to have
died in 1618 until Mark Napier showed that the true date was
1617. When the *Descriptio* was published Briggs was fifty-seven
years of age, and the remaining seventeen years of his
life were devoted with steady enthusiasm to extend the utility
of Napier’s great invention.

The only other mathematician besides Napier who grasped
the idea on which the use of logarithm depends and applied it
to the construction of a table is Justus Byrgius (Jobst Bürgi),
whose work *Arithmetische und geometrische Progress-Tabulen*
... was published at Prague in 1620, six years after the publication
of the *Descriptio* of Napier. This table distinctly involves
the principle of logarithms and may be described as a modified
table of antilogarithms. It consists of two series of numbers,
the one being an arithmetical and the other a geometrical
progression: thus

0, 1,0000 0000 |

10, 1,0001 0000 |

20, 1,0002 0001 |

. . . . |

990, 1,0099 4967 |

. . . . |

In the arithmetical column the numbers increase by 10, in the
geometrical column each number is derived from its predecessor
by multiplication by 1.0001. Thus the number 10*x* in the arithmetical
column corresponds to 10^{8} (1.0001)^{x} in the geometrical
column; the intermediate numbers being obtained by interpolation.
If we divide the numbers in the geometrical column
by 10^{8} the correspondence is between 10*x* and (1.0001)^{x}, and
the table then becomes one of antilogarithms, the base being
(1.0001)^{1/10}, viz. for example (l.0001)^{1/10·990} = 1.00994967. The
table extends to 230270 in the arithmetical column, and it is
shown that 230270.022 corresponds to 9.9999 9999 or 109 in
the geometrical column; this last result showing that
(1.0001)^{23027.022} = 10. The first contemporary mention of Byrgius’s
table occurs on page 11 of the “Praecepta” prefixed to Kepler’s
*Tabulae Radolphinae* (1627); his words are: “apices logistici
J. Byrgio multis annis ante editionem Neperianam viam praeiverent
ad hos ipsissimos logarithmos. Etsi homo cunctator
et secretorum suorum custos foetum in partu destituit, non ad
usus publicos educavit.” Another reference to Byrgius occurs
in a work by Benjamin Bramer, the brother-in-law and pupil
of Byrgius, who, writing in 1630, says that the latter constructed
his table twenty years ago or more.^{[4]}

As regards priority of publication, Napier has the advantage by six years, and even fully accepting Bramer’s statement, there are grounds for believing that Napier’s work dates from a still earlier period.

The power of 10, which occurs as a factor in the tables of both Napier and Byrgius, was rendered necessary by the fact that the decimal point was not yet in use. Omitting this factor in the case of both tables, the connexion between N a number and L its “logarithm” is

*e*

^{−1})

^{L}(Napier), L =(1.0001)

^{1/10N}(Byrgius),

viz. Napier gives logarithms to base *e*^{−1}, Byrgius gives antilogarithms
to base (1.0001)^{1/10}.

There is indirect evidence that Napier was occupied with
logarithms as early as 1594, for in a letter to P. Crügerus
from Kepler, dated September 9, 1624 (Frisch’s *Kepler*, vi. 47),
there occurs the sentence: “Nihil autem supra Neperianam
rationem esse puto: etsi quidem Scotus quidam literis ad
Tychonem 1594 scriptis jam spem fecit Canonis illius Mirifici.”
It is here distinctly stated that some Scotsman in the year 1594,
in a letter to Tycho Brahe, gave him some hope of the logarithms;
and as Kepler joined Tycho after his expulsion from the island
of Huen, and had been so closely associated with him in his
work, he would be likely to be correct in any assertion of this
kind. In connexion with Kepler’s statement the following story,
told by Anthony wood in the *Athenae Oxonienses*, is of some
importance:—

“It must be now known, that one Dr Craig, a Scotchman ...
coming out of Denmark into his own country, called upon Joh.
Neper, Baron of Mercheston, near Edinburgh, and told him, among
other discourses, of a new invention in Denmark (by Longomontanus,
as ’tis said), to save the tedious multiplication and division in astronomical
calculations. Neper being solicitous to know farther of him
concerning this matter, he could give no other account of it than that
it was by proportional numbers. Which hint Neper taking, he
desired him at his return to call upon him again. Craig, after some
weeks had passed, did so, and Neper then showed him a rude draught
of what he called *Canon mirabilis logarithmorum*. which draught,
with some alterations, he printing in 1614, it came forthwith into
the hands of our author Briggs, and into those of Will. Oughtred,
from whom the relation of this matter came.”

This story, though obviously untrue in some respects, gives
valuable information by connecting Dr Craig with Napier and
Longomontanus, who was Tycho Brahe’s assistant. Dr Craig
was John Craig, the third son of Thomas Craig, who was one of the
colleagues of Sir Archibald Napier, John Napier’s father, in the
office of justice-depute. Between John Craig and John Napier a
friendship sprang up which may have been due to their common
taste for mathematics. There are extant three letters from
Dr John Craig to Tycho Brahe, which show that he was on the
most friendly terms with him. In the first letter, of which the
date is not given, Craig says that Sir William Stuart has safely
delivered to him, “about the beginning of last winter,” the book
which he sent him. Now Mark Napier found in the library of
the university of Edinburgh a mathematical work bearing a
sentence in Latin which he translates, “To Doctor John Craig
of Edinburgh, in Scotland, a most illustrious man, highly gifted
with various and excellent learning, professor of medicine, and
exceedingly skilled in the mathematics, Tycho Brahe hath sent
this gift, and with his own hand written this at Uraniburg,
2d November 1588.” As Sir William Stuart was sent to
Denmark to arrange the preliminaries of King James’s marriage,
and returned to Edinburgh on the 15th of November 1588, it
would seem probable that this was the volume referred to by Craig.
It appears from Craig’s letter, to which we may therefore assign
the date 1589, that, five years before, he had made an attempt to
reach Uranienburg, but had been baffled by the storms and rocks
of Norway, and that ever since then he had been longing to visit
Tycho. Now John Craig was physician to the king, and in 1590
James VI. spent some days at Uranienburg, before returning
to Scotland from his matrimonial expedition. It seems not
unlikely therefore that Craig may have accompanied the king
in his visit to Uranienburg.^{[5]} In any case it is certain that
Craig was a friend and correspondent of Tycho’s, and it is probable
that he was the “Scotus quidam.”

We may infer therefore that as early as 1594 Napier had
communicated to some one, probably John Craig, his hope of
being able to effect a simplification in the processes of arithmetic.
Everything tends to show that the invention of logarithms
was the result of many years of labour and thought,^{[6]} undertaken
with this special object, and it would seem that Napier had seen
some prospect of success nearly twenty years before the publication
of the *Descriptio*. It is very evident that no mere hint
with regard to the use of proportional numbers could have been
of any service to him, but it is possible that the news brought
by Craig of the difficulties placed in the progress of astronomy
by the labour of the calculations may have stimulated him to
persevere in his efforts.

The “new invention in Denmark” to which Anthony Wood
refers as having given the hint to Napier was probably the method
of calculation called prosthaphaeresis (often written in Greek
letters προσθαφαίρεσις), which had its origin in the solution of
spherical triangles.^{[7]} The method consists in the use of the
formula

*a*sin

*b*= 12 {cos (

*a*−

*b*) − cos (

*a*+

*b*)},

by means of which the multiplication of two sines is reduced to
the addition or subtraction of two tabular results taken from
a table of sines; and, as such products occur in the solution of
spherical triangles, the method affords the solution of spherical
triangles in certain cases by addition and subtraction only.
It seems to be due to Wittich of Breslau, who was assistant for
a short time to Tycho Brahe; and it was used by them in their
calculations in 1582. Wittich in 1584 made known at Cassel
the calculation of one case by this prosthaphaeresis; and
Justus Byrgius proved it in such a manner that from his proof
the extension to the solution of all triangles could be deduced.^{[8]}
Clavius generalized the method in his treatise *De astrolabio* (1593),
lib. i. lemma liii. The lemma is enunciated as follows:—

“Quaestiones omnes, quae per sinus, tangentes, atque secantes absolvi solent, per solam prosthaphaeresim, id est, per solam additionem, subtractionem, sine laboriosa numerorum multiplicatione divisioneque expedire.”

Clavius then refers to a work of Raymarus Ursus Dithmarsus
as containing an account of a particular case. The work is
probably the *Fundamentum astronomicum* (1588). Longomontanus,
in his *Astronomia Danica* (1622), gives an account of
the method, stating that it is not to be found in the writings
of the Arabs or Regiomontanus. As Longomontanus is mentioned
in Anthony Wood’s anecdote, and as Wittich as well as
Longomontanus were assistants of Tycho, we may infer that
Wittich’s prosthaphaeresis is the method referred to by Wood.

It is evident that Wittich’s prosthaphaeresis could not be
a good method of practically effecting multiplications unless the
quantities to be multiplied were sines, on account of the labour
of the interpolations. It satisfies the condition, however, equally
with logarithms, of enabling multiplication to be performed
by the aid of a table of single entry; and, analytically considered,
it is not so different in principle from the logarithmic method.
In fact, if we put *xy* = φ(X + Y), X being a function of *x* only
and Y a function of *y* only, we can show that we must have
X = A*e*^{qx}, *y* = B*e*^{qy}; and if we put *xy* = φ(X + Y) − φ(X − Y),
the solutions are φ(X + Y) = 14(*x* + *y*)^{2}, and *x* = sin X, *y* = sin Y,
φ(X + Y) = −12 cos(X + Y). The former solution gives a method
known as that of quarter-squares; the latter gives the method
of prosthaphaeresis.

An account has now been given of Napier’s invention and
its publication, the transition to decimal logarithms, the calculation
of the tables by Briggs, Vlacq and Gunter, as well as of
the claims of Byrgius and the method of prosthaphaeresis. To
complete the early history of logarithms it is necessary to return
to Napier’s *Descriptio* in order to describe its reception on the
continent, and to mention the other logarithmic tables which were
published while Briggs was occupied with his calculations.

John Kepler, who has been already quoted in connexion with
Craig’s visit to Tycho Brahe, received the invention of logarithms
almost as enthusiastically as Briggs. His first mention of the
subject occurs in a letter to Schikhart dated the 11th of March
1618, in which he writes-“Extitit Scotus Baro, cujus nomen
mihi excidit, qui praeclari quid praestitit, necessitate omni
multiplicationum et divisionum in meras additiones et subtractiones
commutata, nec sinibus utitur; at tamen opus est
ipsi tangentium canone: et varietas, crebritas, difficultasque
additionum subtractionumque alicubi laborem multiplicandi
et dividendi superat.” This erroneous estimate was formed
when he had seen the *Descriptio* but had not read it; and his
opinion was very different when he became acquainted with the
nature of logarithms. The dedication of his *Ephemeris* for 1620
consists of a letter to Napier dated the 28th of July 1619, and he
there congratulates him warmly on his invention and on the
benefit he has conferred upon astronomy generally and upon
Kepler’s own Rudolphine tables. He says that, although
Napier’s book had been published five years, he first saw it at
Prague two years before; he was then unable to read it, but last
year he had met with a little work by Benjamin Ursinus^{[9]} containing
the substance of the method, and he at once recognized
the importance of what had been effected. He then explains
how he verified the canon, and so found that there were no
essential errors in it, although there were a few inaccuracies
near the beginning of the quadrant, and he proceeds, “Haec
te obiter scire volui, ut quibus tu methodis incesseris, quas non
dubito et plurimas et ingeniosissimas tibi in promptu esse, eas
publici juris fieri, mihi saltem (puto et caeteris) scires fore gratissimum;
eoque percepto, tua promissa folio 57, in debitum
cecidisse intelligeres.” This letter was written two years after
Napier’s death (of which Kepler was unaware), and in the same
year as that in which the *Constructio* was published. In the
same year (1620) Napier’s *Descriptio* (1614) and *Constructio*
(1619) were reprinted by Bartholomew Vincent at Lyons and
issued together.^{[10]}

Napier calculated no logarithms of numbers, and, as already
stated, the logarithms invented by him were not to base *e*.
The first logarithms to the base *e* were published by John Speidell
in his *New Logarithmes* (London, 1619), which contains hyperbolic
log sines, tangents and secants for every minute of the
quadrant to 5 places of decimals.

In 1624 Benjamin Ursinus published at Cologne a canon of
logarithms exactly similar to Napier’s in the *Descriptio* of 1614,
only much enlarged. The interval of the arguments is 10″,
and the results are given to 8 places; in Napier’s canon the
interval is 1′, and the number of places is 7. The logarithms are
strictly Napierian, and the arrangement is identical with that
in the canon of 1614. This is the largest Napierian canon that
has ever been published.

In the same year (1624) Kepler published at Marburg a table of Napierian logarithms of sines with certain additional columns to facilitate special calculations.

The first publication of Briggian logarithms on the continent
is due to Wingate, who published at Paris in 1625 his *Arithmétique*
*logarithmétique*, containing seven-figure logarithms of
numbers up to 1000, and log sines and tangents from Gunter’s
*Canon* (1620). In the following year, 1626, Denis Henrion
published at Paris a *Traicté des Logarithmes*, containing Briggs’s
logarithms of numbers up to 20,001 to 10 places, and Gunter’s
log sines and tangents to 7 places for every minute. In the same
year de Decker also published at Gouda a work entitled *Nieuwe*
*Telkonst, inhoudende de Logarithmi voor de Ghetallen beginnende*
*van 1 tot 10,000*, which contained logarithms of numbers up to
10,000 to 10 places, taken from Briggs’s *Arithmetica* of 1624, and
Gunter’s log sines and tangents to 7 places for every minute.^{[11]}
Vlacq rendered assistance in the publication of this work, and
the privilege is made out to him.

The invention of logarithms and the calculation of the earlier
tables form a very striking episode in the history of exact science,
and, with the exception of the *Principia* of Newton, there is
no mathematical work published in the country which has produced
such important consequences, or to which so much interest
attaches as to Napier’s *Descriptio*. The calculation of tables
of the natural trigonometrical functions may be said to have
formed the work of the last half of the 16th century, and the great
canon of natural sines for every 10 seconds to 15 places which
had been calculated by Rheticus was published by Pitiscus only
in 1613, the year before that in which the *Descriptio* appeared.
In the construction of the natural trigonometrical tables Great
Britain had taken no part, and it is remarkable that the discovery
of the principles and the formation of the tables that were to
revolutionize or supersede all the methods of calculation then
in use should have been so rapidly effected and developed in a
country in which so little attention had been previously devoted
to such questions.

For more detailed information relating to Napier, Briggs and
Vlacq, and the invention of logarithms, the reader is referred to the
life of Briggs in Ward’s *Lives of the Professors of Gresham College*
(London, 1740); Thomas Smith’s *Vitae quorundam eruditissimorum*
*et illustrium virorum* (Vita Henrici Briggii) (London, 1707); Mark
Napier’s *Memoirs of John Napier* already referred to, and the same
author’s *Naperi libri qui supersunt* (1839); Hutton’s *History*; de
Morgan’s article already referred to; Delambre’s *Histoire de l’Astronomie*
*moderne*; the report on mathematical tables in the *Report of*
*the British Association* for 1873; and the *Philosophical Magazine* for
October and December 1872 and May 1873. It may be remarked
that the date usually assigned to Briggs’s first visit to Napier is 1616
and not 1615 as stated above, the reason being that Napier was
generally supposed to have died in 1618; but it was shown by Mark
Napier that the true date is 1617.

In the years 1791–1807 Francis Maseres published at London,
in six volumes quarto “Scriptores Logarithmici, or a collection
of several curious tracts on the nature and construction of
logarithms, mentioned in Dr Hutton’s historical introduction
to his new edition of Sherwin’s mathematical tables ...,”
which contains reprints of Napier’s *Descriptio* of 1614, Kepler’s
writings on logarithms (1624–1625), &c. In 1889 a translation
of Napier’s *Constructio* of 1619 was published by Walter Rae
Macdonald. Some valuable notes are added by the translator,
in one of which he shows the accuracy of the method employed
by Napier in his calculations, and explains the origin of a small
error which occurs in Napier’s table. Appended to the Catalogue
is a full and careful bibliography of all Napier’s writings, with
mention of the public libraries, British and foreign, which possess
copies of each. A facsimile reproduction of Bartholomew
Vincent’s Lyons edition (1620) of the *Constructio* was issued in
1895 by A. Hermann at Paris (this imprint occurs on page 62
after the word “Finis”).

It now remains to notice briefly a few of the more important events in the history of logarithmic tables subsequent to the original calculations.

*Common or Briggian Logarithms of Numbers.*—Nathaniel Roe’s
*Tabulae logarithmicae* (1633) was the first complete seven-figure
table that was published. It contains seven-figure logarithms of
numbers from 1 to 100,000, with characteristics unseparated from the
mantissae, and was formed from Vlacq’s table (1628) by leaving out
the last three figures. All the figures of the number are given at the
head of the columns, except the last two, which run down the
extreme columns—1 to 50 on the left-hand side, and 50 to 100 on the
right-hand side. The first four figures of the logarithms are printed
at the top of the columns. There is thus an advance half way towards
the arrangement now universal in seven-figure tables. The final step
was made by John Newton in his *Trigonometria Britannica* (1658),
a work which is also noticeable as being the only extensive eight-figure
table that until recently had been published; it contains
logarithms of sines, &c., as well as logarithms of numbers.

In 1705 appeared the original edition of Sherwin’s tables, the first of the series of ordinary seven-figure tables of logarithms of numbers and trigonometrical functions such as are in general use now. The work went through several editions during the 18th century, and was at length superseded in 1785 by Hutton’s tables, which continued in successive editions to maintain their position for a century.

In 1717 Abraham Sharp published in his *Geometry Improv’d* the
Briggian logarithms of numbers from 1 to 100, and of primes from
100 to 1100, to 61 places; these were copied into the later editions
of Sherwin and other works.

In 1742 a seven-figure table was published in quarto form by Gardiner, which is celebrated on account of its accuracy and of the elegance of the printing. A French edition, which closely resembles the original, was published at Avignon in 1770.

In 1783 appeared at Paris the first edition of François Callet’s tables, which correspond to those of Hutton in England. These tables, which form perhaps the most complete and practically useful collection of logarithms for the general computer that has been published, passed through many editions.

In 1794 Vega published his *Thesaurus logarithmorum completus*,
a folio volume containing a reprint of the logarithms of numbers
from Vlacq’s *Arithmetica logarithmica* of 1628, and *Trigonometria*
*artificialis* of 1633. The logarithms of numbers are arranged as in
an ordinary seven-figure table. In addition to the logarithms
reprinted from the *Trigonometria*, there are given logarithms for
every second of the first two degrees, which were the result of an
original calculation. Vega devoted great attention to the detection
and correction of the errors in Vlacq’s work of 1628. Vega’s *Thesaurus*
has been reproduced photographically by the Italian government.
Vega also published in 1797, in 2 vols. 8vo, a collection of logarithmic
and trigonometrical tables which has passed through many editions,
a very useful one volume stereotype edition having been published in
1840 by Hülsse. The tables in this work may be regarded as to some
extent supplementary to those in Callet.

If we consider only the logarithms of numbers, the main line of
descent from the original calculation of Briggs and Vlacq is Roe,
John Newton, Sherwin, Gardiner; there are then two branches,
viz. Hutton founded on Sherwin and Callet on Gardiner, and the
editions of Vega form a separate offshoot from the original tables.
Among the most useful and accessible of modern ordinary seven-figure
tables of logarithms of numbers and trigonometrical functions
may be mentioned those of Bremiker, Schrön and Bruhns. For
logarithms of numbers only perhaps Babbage’s table is the most
convenient.^{[12]}

In 1871 Edward Sang published a seven-figure table of logarithms
of numbers from 20,000 to 200,000, the logarithms between 100,000
and 200,000 being the result of a new calculation. By beginning the
table at 20,000 instead of at 10,000 the differences are halved in
magnitude, while the number of them in a page is quartered. In this
table multiples of the differences, instead of proportional parts, are
given.^{[13]} John Thomson of Greenock (1782–1855) made an independent
calculation of logarithms of numbers up to 120,000 to 12
places of decimals, and his table has been used to verify the errata
already found in Vlacq and Briggs by Lefort (see *Monthly Not. R.A.S.*
vol. 34, p. 447). A table of ten-figure logarithms of numbers up to
100,009 was calculated by W. W. Duffield and published in the
*Report of the U.S. Coast and Geodetic Survey for 1895–1896* as Appendix
12, pp. 395–722. The results were compared with Vega’s *Thesaurus*
(1794) before publication.

*Common or Briggian Logarithms of Trigonometrical Functions.*—The
next great advance on the *Trigonometria artificialis* took place
more than a century and a half afterwards, when Michael Taylor
published in 1792 his seven-decimal table of log sines and tangents
to every second of the quadrant; it was calculated by interpolation
from the *Trigonometria* to 10 places and then contracted to 7. On
account of the great size of this table, and for other reasons, it never
came into very general use, Bagay’s *Nouvelles tables astronomiques*
(1829), which also contains log sines and tangents to every second,
being preferred; this latter work, which for many years was difficult
to procure, has been reprinted with the original title-page and date
unchanged. The only other logarithmic canon to every second that
has been published forms the second volume of Shortrede’s *Logarithmic*
*Tables* (1849). In 1784 the French government decided that
new tables of sines, tangents, &c., and their logarithms, should be
calculated in relation to the centesimal division of the quadrant.
Prony was charged with the direction of the work, and was expressly
required “non seulement à composer des tables qui ne laissassent rien
à désirer quant à l’exactitude, mais à en faire le monument de calcul
le plus vaste et le plus imposant qui eût jamais été exécuté ou même
conçu.” Those engaged upon the work were divided into three
sections: the first consisted of five or six mathematicians, including
Legendre, who were engaged in the purely analytical work, or the
calculation of the fundamental numbers; the second section consisted
of seven or eight calculators possessing some mathematical
knowledge; and the third comprised seventy or eighty ordinary
computers. The work, which was performed wholly in duplicate,
and independently by two divisions of computers, occupied two years.
As a consequence of the double calculation, there are two manuscripts,
one deposited at the Observatory, and the other in the library of the
Institute, at Paris. Each of the two manuscripts consists essentially
of seventeen large folio volumes, the contents being as follows:—

Logarithms of numbers up to 200,000 | 8 | vols. |

Natural sines | 1 | ” |

Logarithms of the ratios of arcs to sines from 0^{q}.00000 | ||

to 0^{q}.05000, and log sines throughout the quadrant | 4 | ” |

Logarithms of the ratios of arcs to tangents from | ||

0^{q}.00000 to 0^{q}.05000, and log tangents throughout | ||

the quadrant | 4 | ” |

The trigonometrical results are given for every hundred-thousandth
of the quadrant (10″ centesimal or 3″.24 sexagesimal). The tables
were all calculated to 14 places, with the intention that only 12
should be published, but the twelfth figure is not to be relied upon.
The tables have never been published, and are generally known as the
*Tables du Cadastre*, or, in England, as the great French manuscript
tables.

A very full account of these tables, with an explanation of the
methods of calculation, formulae employed, &c., was published by
Lefort in vol. iv. of the *Annales de l’observatoire de Paris*. The printing
of the table of natural sines was once begun, and Lefort states
that he has seen six copies, all incomplete, although including the
last page. Babbage compared his table with the *Tables du Cadastre*,
and Lefort has given in his paper just referred to most important
lists of errors in Vlacq’s and Briggs’s logarithms of numbers which
were obtained by comparing the manuscript tables with those contained
in the *Arithmetica logarithmica* of 1624 and of 1628.

As the *Tables du Cadastre* remained unpublished, other tables
appeared in which the quadrant was divided centesimally, the most
important of these being Hobert and Ideler’s *Nouvelles tables trigonométriques*
(1799), and Borda and Delambre’s *Tables trigonométriques*
*décimales* (1800–1801), both of which are seven-figure tables. The
latter work, which was much used, being difficult to procure, and
greater accuracy being required, the French government in 1891
published an eight-figure centesimal table, for every ten seconds,
derived from the *Tables du Cadastre*.

*Decimal or Briggian Antilogarithms.*—In the ordinary tables of
logarithms the natural numbers are all integers, while the logarithms
tabulated are incommensurable. In an antilogarithmic table, the
logarithms are exact quantities such as .00001, .00002, &c., and the
numbers are incommensurable. The earliest and largest table of
this kind that has been constructed is Dodson’s *Antilogarithmic canon*
(1742), which gives the numbers to 11 places, corresponding to the
logarithms from .00001 to .99999 at intervals of .00001. Antilogarithmic
tables are few in number, the only other extensive tables of
the same kind that have been published occurring in Shortrede’s
*Logarithmic tables* already referred to, and in Filipowski’s *Table of*
*antilogarithms* (1849). Both are similar to Dodson’s tables, from
which they were derived, but they only give numbers to 7 places.

*Hyperbolic or Napierian logarithms* (*i.e.* to base *e*).—The most
elaborate table of hyperbolic logarithms that exists is due to Wolfram,
a Dutch lieutenant of artillery. His table gives the logarithms of all
numbers up to 2200, and of primes (and also of a great many composite
numbers) from 2200 to 10,009, to 48 decimal places. The table
appeared in Schulze’s *Neue und erweiterte Sammlung logarithmischer*
*Tafeln* (1778), and was reprinted in Vega’s *Thesaurus* (1794), already
referred to. Six logarithms omitted in Schulze’s work, and which
Wolfram had been prevented from computing by a serious illness,
were published subsequently, and the table as given by Vega is
complete. The largest hyperbolic table as regards range was
published by Zacharias Dase at Vienna in 1850 under the title *Tafel*
*der natürlichen Logarithmen der Zahlen*.

*Hyperbolic antilogarithms* are simple exponentials, *i.e.* the hyperbolic
antilogarithm of *x* is *e*^{x}. Such tables can scarcely be said to
come under the head of logarithmic tables. See Tables, Mathematical:
*Exponential Functions*.

*Logistic or Proportional Logarithms.*—The old name for what are
now called ratios or fractions are *logistic numbers*, so that a table of
log (*a*/*x*) where *x* is the argument and a a constant is called a table of
logistic or proportional logarithms; and since log (*a*/*x*) = log *a* − log *x*
it is clear that the tabular results differ from those given in an ordinary
table of logarithms only by the subtraction of a constant and a
change of sign. The first table of this kind appeared in Kepler’s
work of 1624 which has been already referred to. The object of a
table of log (*a*/*x*) is to facilitate the working out of proportions in
which the third term is a constant quantity *a*. In most collections
of tables of logarithms, and especially those intended for use in
connexion with navigation, there occurs a small table of logistic
logarithms in which *a* = 3600″ (= 1° or 1^{h}), the table giving log 3600 − log *x*,
and *x* being expressed in minutes and seconds. It is also
common to find tables in which *a* = 10800″ (= 3° or 3^{h}), and *x* is expressed
in degrees (or hours), minutes and seconds. Such tables are
generally given to 4 or 5 places. The usual practice in books seems
to be to call logarithms logistic when a is 3600″, and proportional
when a has any other value.

*Addition and Subtraction, or Gaussian Logarithms.*—*Gaussian*
*logarithms* are intended to facilitate the finding of the logarithms of
the sum and difference of two numbers whose logarithms are known,
the numbers themselves being unknown; and on this account they
are frequently called addition and subtraction logarithms. The
object of the table is in fact to give log (*a* ± *b*) by only one entry when
log a and log *b* are given. The utility of such logarithms was first
pointed out by Leonelli in a book entitled *Supplément logarithmique*,
printed at Bordeaux in the year XI. (1802/3); he calculated a
table to 14 places, but only a specimen of it which appeared in the
*Supplément* was printed. The first table that was actually published
is due to Gauss, and was printed in Zach’s *Monatliche Correspondenz*,
xxvi. 498 (1812). Corresponding to the argument log *x* it gives
the values of log (1 + *x*^{−1}) and log (1 + *x*).

*Dual Logarithms.*—This term was used by Oliver Byrne in a series
of works published between 1860 and 1870. Dual numbers and
logarithms depend upon the expression of a number as a product of
1.1, 1.01, 1.001 ... or of .9, .99, .999....

In the preceding *résumé* only those publications have been
mentioned which are of historic importance or interest.^{[14]} For fuller
details with respect to some of these works, for an account of tables
published in the latter part of the 19th century, and for those which
would now be used in actual calculation, reference should be made
to the article Tables, Mathematical.

*Calculation of Logarithms.*—The name logarithm is derived from
the words λόγων ἀριθμός, the number of the ratios, and the way of
regarding a logarithm which justifies the name may be explained as
follows. Suppose that the ratio of 10, or any other particular number,
to 1 is compounded of a very great number of equal ratios, as, for
example, 1,000,000, then it can be shown that the ratio of 2 to 1 is
very nearly equal to a ratio compounded of 301,030 of these small
ratios, or *ratiunculae*, that the ratio of 3 to 1 is very nearly equal
to a ratio compounded of 477,121 of them, and so on. The small
ratio, or *ratiuncula*, is in fact that of the millionth root of 10 to unity,
and if we denote it by the ratio of a to 1, then the ratio of 2 to 1 will
be nearly the same as that of a301,030 to 1, and so on; or, in other
words, if a denotes the millionth root of 10, then 2 will be nearly
equal to a301,030, 3 will be nearly equal to a477,121, and so on.

Napier’s original work, the *Descriptio Canonis* of 1614, contained,
not logarithms of numbers, but logarithms of sines, and the relations
between the sines and the logarithms were explained by the motions
of points in lines, in a manner not unlike that afterwards employed
by Newton in the method of fluxions. An account of the processes
by which Napier constructed his table was given in the *Constructio*
*Canonis* of 1619. These methods apply, however, specially to
Napier’s own kind of logarithms, and are different from those actually
used by Briggs in the construction of the tables in the *Arithmetica*
*Logarithmica*, although some of the latter are the same in principle
as the processes described in an appendix to the *Constructio*.

The processes used by Briggs are explained by him in the preface
to the *Arithmetica Logarithmica* (1624). His method of finding the
logarithms of the small primes, which consists in taking a great
number of continued geometric means between unity and the given
primes, may be described as follows. He first formed the table of
numbers and their logarithms:—

Numbers. | Logarithms, |

10 | 1 |

3.162277... | 0.5 |

1.778279... | 0.25 |

1.333521... | 0.125 |

1.154781... | 0.0625 |

each quantity in the left-hand column being the square root of the one
above it, and each quantity in the right-hand column being the half
of the one above it. To construct this table Briggs, using about
thirty places of decimals, extracted the square root of 10 fifty-four
times, and thus found that the logarithm of 1.00000 00000 00000
12781 91493 20032 35 was 0.00000 00000 00000 05551 11512 31257
82702, and that for numbers of this form (*i.e.* for numbers beginning
with 1 followed by fifteen ciphers, and then by seventeen or a less
number of significant figures) the logarithms were proportional to
these significant figures. He then by means of a simple proportion
deduced that log (1.00000 00000 00000 1) = 0.00000 00000 00000
04342 94481 90325 1804, so that, a quantity 1.00000 00000 00000 x
(where *x* consists of not more than seventeen figures) having been
obtained by repeated extraction of the square root of a given number,
the logarithm of 1.00000 00000 00000 *x* could then be found by
multiplying *x* by .00000 00000 00000 04342....

To find the logarithm of 2, Briggs raised it to the tenth power, viz.
1024, and extracted the square root of 1.024 forty-seven times, the
result being 1.00000 00000 00000 16851 60570 53949 77. Multiplying
the significant figures by 4342 ... he obtained the logarithm of this
quantity, viz. 0.00000 00000 00000 07318 55936 90623 9336, which
multiplied by 2^{47} gave 0.01029 99566 39811 95265 277444, the
logarithm of 1.024, true to 17 or 18 places. Adding the characteristic
3, and dividing by 10, he found (since 2 is the tenth root of 1024)
log 2 = .30102 99956 63981 195. Briggs calculated in a similar
manner log 6, and thence deduced log 3.

It will be observed that in the first process the value of the modulus is in fact calculated from the formula.

h |
= | 1 | , |

10^{h} − 1 | log_{e} 10 |

the value of *h* being 1/2^{54}, and in the second process log_{10} 2 is in effect
calculated from the formula.

log_{10} 2 = ( 2102^{47} − 1 ) × | 1 | × | 2^{47} |
. |

log_{e} 10 | 10 |

Briggs also gave methods of forming the mean proportionals or square roots by differences; and the general method of constructing logarithmic tables by means of differences is due to him.

The following calculation of log 5 is given as an example of the
application of a method of mean proportionals. The process consists
in taking the geometric mean of numbers above and below 5, the
object being to at length arrive at 5.000000. To every geometric
mean in the column of numbers there corresponds the arithmetical
mean in the column of logarithms. The numbers are denoted by
*A*, *B*, *C*, &c., in order to indicate their mode of formation.

Numbers. | Logarithms. | ||

A = | 1.000000 | 0.0000000 | |

B = | 10.000000 | 1.0000000 | |

C = √(AB) | = | 3.162277 | 0.5000000 |

D = √(BC) | = | 5.623413 | 0.7500000 |

E = √(CD) | = | 4.216964 | 0.6250000 |

F = √(DE) | = | 4.869674 | 0.6875000 |

G = √(DF) | = | 5.232991 | 0.7187500 |

H = √(FG) | = | 5.048065 | 0.7031250 |

I =√(FH) | = | 4.958069 | 0.6953125 |

K = √(HI) | = | 5.002865 | 0.6992187 |

L =√(IK) | = | 4.980416 | 0.6972656 |

M = √(KL) | = | 4.991627 | 0.6982421 |

N = √(KM) | = | 4.997242 | 0.6987304 |

O = √(KN) | = | 5.000052 | 0.6989745 |

P = √(NO) | = | 4.998647 | 0.6988525 |

Q = √(OP) | = | 4.999350 | 0.6989135 |

R = √(OQ) | = | 4.999701 | 0.6989440 |

S = √(OR) | = | 4.999876 | 0.6989592 |

T = √(OS) | = | 4.999963 | 0.6989668 |

V = √(OT) | = | 5.000008 | 0.6989707 |

W =√(TV) | = | 4.999984 | 0.6989687 |

X = √(WV) | = | 4.999997 | 0.6989697 |

Y = √(VX) | = | 5.000003 | 0.6989702 |

Z = √(XY) | = | 5.000000 | 0.6989700 |

Great attention was devoted to the methods of calculating
logarithms during the 17th and 18th centuries. The earlier methods
proposed were, like those of Briggs, purely arithmetical, and for a
long time logarithms were regarded from the point of view indicated
by their name, that is to say, as depending on the theory of compounded
ratios. The introduction of infinite series into mathematics
effected a great change in the modes of calculation and the treatment
of the subject. Besides Napier and Briggs, special reference should
be made to Kepler (*Chilias*, 1624) and Mercator (*Logarithmotechnia*,
1668), whose methods were arithmetical, and to Newton, Gregory,
Halley and Cotes, who employed series. A full and valuable account
of these methods is given in Hutton’s “Construction of Logarithms,”
which occurs in the introduction to the early editions of his *Mathematical*
*Tables*, and also forms tract 21 of his *Mathematical Tracts*
(vol. i., 1812). Many of the early works on logarithms were reprinted
in the *Scriptores logarithmici* of Baron Maseres already
referred to.

In the following account only those formulae and methods will be referred to which would now be used in the calculation of logarithms.

Since

_{e}(1 +

*x*) =

*x*− 12

*x*

^{2}+ 13

*x*

^{3}− 14

*x*

^{4}+ &c.,

we have, by changing the sign of *x*,

_{e}(1 −

*x*) = −x − 12

*x*

^{2}− 13

*x*

^{3}− 14

*x*

^{4}− &c.;

whence

log_{e} | 1 + x |
= 2 (x + 13x^{3} + 15x^{5} + &c.), |

1 − x |

and, therefore, replacing *x* by (*p* − *q*)/(*p* + *q*),

log_{e} | p |
= 2 { | p − q |
+ 13 ( | p − q |
) 3 + 15 ( | p − q |
) 5 + &c. }, |

q | p + q |
p + q |
p + q |

in which the series is always convergent, so that the formula affords a method of deducing the logarithm of one number from that of another.

As particular cases we have, by putting *q* = 1,

log_{e} p = 2 { | p − 1 |
+ 13 ( | p − 1 |
) 3 + 15 ( | p − 1 |
) 5 + &c. }, |

p + 1 | p + 1 |
p + 1 |

and by putting *q* = *p* + 1,

log_{e}(p + 1) − log_{e} p = 2 { | 1 | + 13 | 1 | + 15 | 1 | + &c. }; |

2p + 1 | (2p + 1)^{3} |
(2p + 1)^{5} |

the former of these equations gives a convergent series for log_{e}*p*, and
the latter a very convergent series by means of which the logarithm
of any number may be deduced from the logarithm of the preceding
number.

From the formula for log_{e} (*p*/*q*) we may deduce the following very
convergent series for log_{e}2, log_{e}3 and log_{e}5, viz.:—

log |

where

P = | 1 | + 13 · | 1 | + 15 · | 1 | + &c. |

31 | (31)^{3} |
(31)^{5} |

Q = | 1 | + 13 · | 1 | + 15 · | 1 | + &c. |

49 | (49)^{3} |
(49)^{5} |

R = | 1 | + 13 · | 1 | + 15 · | 1 | + &c. |

161 | (161)^{3} |
(161)^{5} |

The following still more convenient formulae for the calculation
of log_{e} 2, log_{e} 3, &c. were given by J. Couch Adams in the *Proc. Roy.*
*Soc.*, 1878, 27, p. 91. If

a = log | 10 | = −log ( 1 − | 1 | ), b = log | 25 | = −log ( 1 − | 4 | ), |

9 | 10 | 24 | 100 |

c = log | 81 | = log ( 1 + | 1 | ), d = log | 50 | = −log ( 1 − | 2 | ), |

80 | 80 | 49 | 100 |

e = log | 126 | = log ( 1 + | 8 | ), |

125 | 1000 |

then

*a*− 2

*b*+ 3

*c*, log 3 = 11

*a*− 3

*b*+ 5

*c*, log 5 = 16

*a*− 4

*b*+ 7

*c*,

and

*a*− 10

*b*+ 17

*c*−

*d*) or = 19

*a*− 4

*b*+ 8

*c*+

*e*,

and we have the equation of condition,

*a*− 2

*b*+

*c*=

*d*+ 2

*e*.

By means of these formulae Adams calculated the values of log_{e} 2,
log_{e} 3, log_{e} 5, and log_{e} 7 to 276 places of decimals, and deduced the
value of log_{e} 10 and its reciprocal M, the modulus of the Briggian
system of logarithms. The value of the modulus found by Adams is

Mo = 0.43429 | 44819 | 03251 | 82765 | 11289 |

18916 | 60508 | 22943 | 97005 | 80366 |

65661 | 14453 | 78316 | 58646 | 49208 |

87077 | 47292 | 24949 | 33843 | 17483 |

18706 | 10674 | 47663 | 03733 | 64167 |

92871 | 58963 | 90656 | 92210 | 64662 |

81226 | 58521 | 27086 | 56867 | 03295 |

93370 | 86965 | 88266 | 88331 | 16360 |

77384 | 90514 | 28443 | 48666 | 76864 |

65860 | 85135 | 56148 | 21234 | 87653 |

43543 | 43573 | 17253 | 83562 | 21868 |

25 |

which is true certainly to 272, and probably to 273, places (*Proc. Roy.*
*Soc.*, 1886, 42, p. 22, where also the values of the other logarithms
are given).

If the logarithms are to be Briggian all the series in the preceding formulae must be multiplied by M, the modulus; thus,

_{10}(1 +

*x*) = M (

*x*− 12

*x*

^{2}+ 13

*x*

^{3}− 14

*x*

^{4}+ &c.),

and so on.

As has been stated, Abraham Sharp’s table contains 61-decimal Briggian logarithms of primes up to 1100, so that the logarithms
of all composite numbers whose greatest prime factor does not exceed
this number may be found by simple addition; and Wolfram’s
table gives 48-decimal hyperbolic logarithms of primes up to 10,009.
By means of these tables and of a factor table we may very readily
obtain the Briggian logarithm of a number to 61 or a less number
of places or of its hyperbolic logarithm to 48 or a less number of
places in the following manner. Suppose the hyperbolic logarithm
of the prime number 43,867 required. Multiplying by 50, we have
50 × 43,867 = 2,193,350, and on looking in Burckhardt’s *Table des*
*diviseurs* for a number near to this which shall have no prime factor
greater than 10,009, it appears that

thus

and therefore

_{e}43,867 = log

_{e}23 + log

_{e}47 + log

_{e}2029 − log

_{e}50

+ | 1 | − 12 | 1 | + 13 | 1 | − &c. |

2,193,349 | (2,193,349)^{2} |
(2,193,349)^{3} |

The first term of the series in the second line is

dividing this by 2 × 2,193,349 we obtain

and the third term is

so that the series =

whence, taking out the logarithms from Wolfram’s table,

_{e}43,867 = 10.68891 76079 60568 10191 3661.

The principle of the method is to multiply the given prime (supposed to consist of 4, 5 or 6 figures) by such a factor that the product may be a number within the range of the factor tables, and such that, when it is increased by 1 or 2, the prime factors may all be within the range of the logarithmic tables. The logarithm is then obtained by use of the formula

log_{e} (x + d) = log_{e} x + | d |
− 12 | d^{2} |
+ 13 | d^{3} |
− &c., |

x | x^{2} |
x^{3} |

in which of course the object is to render *d*/*x* as small as possible.
If the logarithm required is Briggian, the value of the series is to
be multiplied by M.

If the number is incommensurable or consists of more than seven
figures, we can take the first seven figures of it (or multiply and
divide the result by any factor, and take the first seven figures of
the result) and proceed as before. An application to the hyperbolic
logarithm of π is given by Burckhardt in the introduction to his
*Table des diviseurs* for the second million.

The best general method of calculating logarithms consists, in its
simplest form, in resolving the number whose logarithm is required
into factors of the form 1 − .1^{r}*n*, where *n* is one of the nine digits;
and making use of subsidiary tables of logarithms of factors of this
form. For example, suppose the logarithm of 543839 required to
twelve places. Dividing by 10^{5} and by 5 the number becomes
1.087678, and resolving this number into factors of the form 1 − .1^{r}*n*
we find that

543839 = 10^{5} | × 5(1 − .1^{2}8) (1 − .1^{4}6) (1 − .1^{5}6) (1 − .1^{6}3) (1 − .1^{7}3) |

× (1 − .1^{8}5) (1 − .1^{9}7) (1 − .1^{10}9) (1 − .1^{11}3) (1 − .1^{12}2), |

where 1 − 1^{2}8 denotes 1 − .08, 1 − .1^{4}6 denotes 1 − .0006, &c., and so
on. All that is required therefore in order to obtain the logarithm
of any number is a table of logarithms, to the required number of
places, of .n, .9n, .99n, .999n, &c., for *n* = 1, 2, 3, ... 9.

The resolution of a number into factors of the above form is easily
performed. Taking, for example, the number 1.087678, the object is
to destroy the significant figure 8 in the second place of decimals;
this is effected by multiplying the number by 1-.08, that is, by
subtracting from the number eight times itself advanced two places,
and we thus obtain 1.00066376. To destroy the first 6 multiply
by 1 − .0006 giving 1.000063361744, and multiplying successively
by 1 − .00006 and 1 − .000003, we obtain 1.000000357932, and it is
clear that these last six significant figures represent without any
further work the remaining factors required. In the corresponding
antilogarithmic process the number is expressed as a product of
factors of the form 1 + .1^{n}x.

This method of calculating logarithms by the resolution of numbers
into factors of the form 1 − .1^{r}*n* is generally known as Weddle’s
method, having been published by him in *The Mathematician* for
November 1845, and the corresponding method for antilogarithms
by means of factors of the form 1 + (.1)^{r}*n* is known by the name of
Hearn, who published it in the same journal for 1847. In 1846 Peter
Gray constructed a new table to 12 places, in which the factors were
of the form 1 − (.01)^{r}*n*, so that *n* had the values 1, 2, ... 99; and
subsequently he constructed a similar table for factors of the form
1 + (.01)^{r}*n*. He also devised a method of applying a table of Hearn’s
form (*i.e.* of factors of the form 1 + .1^{r}*n*) to the construction of
logarithms, and calculated a table of logarithms of factors of the form
1 + (.001)^{r}*n* to 24 places. This was published in 1876 under the title
*Tables for the formation of logarithms and antilogarithms to twenty-four*
*or any less number of places*, and contains the most complete and
useful application of the method, with many improvements in points
of detail. Taking as an example the calculation of the Briggian
logarithm of the number 43,867, whose hyperbolic logarithm has
been calculated above, we multiply it by 3, giving 131,601, and find
by Gray’s process that the factors of 1.31601 are

(1) 1.316 | (5) 1.(001)^{4}002 |

(2) 1.000007 | (6) 1.(001)^{5}602 |

(3) 1.(001)^{2}598 | (7) 1.(001)^{6}412 |

(4) 1.(001)^{3}780 | (8) 1.(001)^{7}340 |

Taking the logarithms from Gray’s tables we obtain the required logarithm by addition as follows:—

522 | 878 | 745 | 280 | 337 | 562 | 704 | 972 = colog 3 |

119 | 255 | 889 | 277 | 936 | 685 | 553 | 913 = log (1) |

3 | 040 | 050 | 733 | 157 | 610 | 239 = log (2) | |

259 | 708 | 022 | 525 | 453 | 597 = log (3) | ||

338 | 749 | 695 | 752 | 424 = log (4) | |||

868 | 588 | 964 = log (5) | |||||

261 | 445 | 278 = log (6) | |||||

178 | 929 = log (7) | ||||||

148 = log (8) | |||||||

4.642 | 137 | 934 | 655 | 780 | 757 | 288 | 464 = log_{10} 43,867 |

In Shortrede’s *Tables* there are tables of logarithms and factors of
the form 1 ± (.01)^{r} *n* to 16 places and of the form 1 ± (.1)^{r} *n* to 25
places; and in his *Tables de Logarithmes à 27 Décimales* (Paris, 1867)
Fédor Thoman gives tables of logarithms of factors of the form
1 ± .1^{r} n. In the *Messenger of Mathematics*, vol. iii. pp. 66-92, 1873,
Henry Wace gave a simple and clear account of both the logarithmic
and antilogarithmic processes, with tables of both Briggian and
hyperbolic logarithms of factors of the form 1 ± .1^{r}*n* to 20 places.

Although the method is usually known by the names of Weddle
and Hearn, it is really, in its essential features, due to Briggs, who
gave in the *Arithmetica logarithmica* of 1624 a table of the logarithms
of 1 + .1^{r}*n* up to *r* = 9 to 15 places of decimals. It was first formally
proposed as an independent method, with great improvements, by
Robert Flower in *The Radix*, *a new way of making Logarithms*, which
was published in 1771; and Leonelli, in his *Supplement logarithmique*
(1802–1803), already noticed, referred to Flower and reproduced
some of his tables. A complete bibliography of this method has been
given by A. J. Ellis in a paper “on the potential radix as a means of
calculating logarithms,” printed in the *Proceedings of the Royal*
*Society*, vol. xxxi., 1881, pp. 401–407, and vol. xxxii., 1881, pp. 377–379.
Reference should also be made to Hoppe’s *Tafeln zur dreissigstelligen*
*logarithmischen Rechnung* (Leipzig, 1876), which give in a
somewhat modified form a table of the hyperbolic logarithm of
1 + .1^{r}*n*.

The preceding methods are only appropriate for the calculation of
isolated logarithms. If a complete table had to be reconstructed, or
calculated to more places, it would undoubtedly be most convenient
to employ the method of differences. A full account of this method
as applied to the calculation of the *Tables du Cadastre* is given by
Lefort in vol. iv. of the *Annales de l’Observatoire de Paris*.
(J. W. L. G.)

- ↑ Dr Thomas Smith thus describes the ardour with which Briggs
studied the
*Descriptio*: “Hunc in deliciis habuit, in sinu, in manibus, in pectore gestavit, oculisque avidissimis, et mente attentissima, iterum iterumque perlegit,...”*Vitae quorundam eruditissimorum et**illustrium virorum*(London, 1707). - ↑ William Lilly’s account of the meeting of Napier and Briggs at Merchiston is quoted in the article Napier.
- ↑ It was certainly published after Napier’s death, as Briggs
mentions his “librum posthumum.” This
*liber posthumus*was the*Constructio*referred to later in this article. - ↑ Frisch’s
*Kepleri opera omnia*, ii. 834. Frisch thinks Bramer possibly relied on Kepler’s statement quoted in the text (“Quibus forte confisus Kepleri verbis Benj. Bramer....”). See also vol. vii. p. 298.

The claims of Byrgius are discussed in Kästner’s*Geschichte der**Mathematik*, ii. 375, and iii. 14; Montucla’s*Histoire des mathématiques*, ii. 10; Delambre’s*Histoire de l’astronomie moderne*, i. 560; de Morgan’s article on “Tables” in the*English**Cyclopaedia*; Mark Napier’s*Memoirs of John Napier of Merchiston*(1834), p. 392, and Cantor’s*Geschichte der Mathematik*, ii. (1892), 662. See also Gieswald,*Justus Byrg als Mathematiker und dessen**Einleitung in seine Logarithmen*(Danzig, 1856). - ↑ See Mark Napier’s
*Memoirs of John Napier of Merchiston*(1834), p. 362. - ↑ In the
*Rabdologia*(1617) he speaks of the canon of logarithms as “a me longo tempore elaboratum.” - ↑ A careful examination of the history of the method is given by
Scheibel in his
*Einleitung zur mathematischen Bücherkenntniss*, Stück vii. (Breslau, 1775), pp. 13-20; and there is also an account in Kästner’s*Geschichte der Mathematik*, i. 566–569 (1796); in Montucla’s*Histoire des mathématiques*, i. 583–585 and 617–619; and in Klügel’s*Wörterbuch*(1808), article “Prosthaphaeresis.” - ↑ Besides his connexion with logarithms and improvements in the
method of prosthaphaeresis, Byrgius has a share in the invention
of decimal fractions. See Cantor,
*Geschichte*, ii. 567. Cantor attributes to him (in the use of his prosthaphaeresis) the first introduction of a subsidiary angle into trigonometry (vol. ii. 590). - ↑ The title of this work is—
*Benjaminis Ursini*...*cursus mathematici**practici volumen primum continens illustr. & generosi Dn.**Dn. Johannis Neperi Baronis Merchistonij &c. Scoti trigonometriam**logarithmicam usibus discentium accommodatam*...*Coloniae*...*CIƆ IƆC XIX*. At the end, Napier’s table is reprinted, but to two figures less. This work forms the earliest publication of logarithms on the continent. - ↑ The title is
*Logarithmorum canonis descriptio, seu arithmeticarum**supputationum mirabilis abbreviatio*.*Ejusque usus in**utraque trigonometria ut etiam in omni logistica mathematica,**amplissimi, facillimi & expeditissimi explicatio. Authore ac inventore**Ioanne Nepero, Barone Merchistonii, &c. Scoto. Lugduni*.... It will be seen that this title is different from that of Napier’s work of 1614; many writers have, however, erroneously given it as the title of the latter. - ↑ In describing the contents of the works referred to, the language
and notation of the present day have been adopted, so that for
example a table to radius 10,000,000 is described as a table to 7
places, and so on. Also, although logarithms have been spoken of as
to the base
*e*, &c., it is to be noticed that neither Napier nor Briggs, nor any of their successors till long afterwards, had any idea of connecting logarithms with exponents. - ↑ The smallest number of entries which are necessary in a table of
logarithms in order that the intermediate logarithms may be calculable
by proportional parts has been investigated by J. E. A. Steggall
in the
*Proc. Edin. Math. Soc.*, 1892, 10, p. 35. This number is 1700 in the case of a seven-figure table extending to 100,000. - ↑ Accounts of Sang’s calculations are given in the
*Trans. Roy. Soc.**Edin.*, 1872, 26, p. 521, and in subsequent papers in the*Proceedings*of the same society. - ↑ In vol. xv. (1875) of the
*Verhandelingen*of the Amsterdam Academy of Sciences, Bierens de Haan has given a list of 553 tables of logarithms. A previous paper of the same kind, containing notices of some of the tables, was published by him in the*Verslagen en**Mededeelingen*of the same academy (Afd. Natuurkunde) deel. iv. (1862), p. 15.