Encyclopaedia Britannica, or, a Dictionary of arts, sciences, and miscellaneous literature : enlarged and improved. Illustrated with nearly six hundred engravings > Volume 13, MAT-MIC

MAT HE
her', of depriving equations of the fecond term, and of
freeing them from fra&ional coefficients. The method
which he has given for refolving equations of the third
and fourth degree is alfo new and ingenious, and his
mode of obtaining an approximate folution of equations
of every order is entitled to ftill higher praife. We
are alfo indebted to Vieta for the theory of angular
fe&ions, the objeft of which is to find the general ex-
preffions of the chords or fines for a feries of arcs that
are multiples of each other.
Logarithms 27. While analyfis was making fuch progrefs on the
invented by continent, Baron Napier of Merchifton in Scotland was
Baron !N a- krjncrinnr to perfeftion his illuftrious difcovery of the lo-
Born 1550. garithms, a fet of artificial numbers, by which the moft
Died 1617. tedious operations in multiplication and divifion may be
performed merely by addition and fubtraftion. This
difcovery was publiffied at Edinburgh in 1614 in his
work entitled Logarithmorum Canonis Defcriptio, feu
Arithmetica Supputationum Mirabilis Abbreviatio. It
is well known that there is fuch a correfpondence be¬
tween every arithmetical and geometrical progreffion,
viz. q’ i’ \ that any terms of
L 2, 4, 8, 16, 32, 64, J ^
the geometrical progreffion may be multiplied or divided
by merely adding or fubtrafting the correfponding
terms of the arithmetical progreffion ; thus the produft
of four and eight may be found by taking the fum of
the correfponding terms in the arithmetical progreffion,
viz. 2 and 3, for their fum 5 points out 32 as the pro
dud of 4 and 8. The numbers o, I, 2, 3, &c. are
therefore the logarithms of 1, 2, 4, 8, &c. The
choice of the two progreffions being altogether arbi¬
trary, Baron Napier took the arithmetical progreffion
which we have given above, and made the term o cor-
refpond with the unit of the geometrical progreffion,,
which he regulated in fuch a manner that when its
terms are reprefented by the abfciffge of an equilateral
hyperbola in which the firft abfcifs and the firft ordi¬
nate are each equal to 1, the logarithms are reprefent-
Sfables of ed by the hyperbolic fpaces. In confequence, however,
logarithms cf t^e inconvenience of this geometrical progreffion,
by Mr ^ Baron Napier, after confulting upon the fubjeft with
Briggs. Henry Briggs of Greffiam College, fubftituted the de¬
cuple progreffion 1, 10, ico, 1000, of which o, 1, 2,
3, 4, &c. are the logarithms. Nothing now remained
but to conftruft tables of logarithms, by finding the lo¬
garithms of the intermediate numbers between the terms
A.D. 1618. of the decuple progreffion. Napier, however, died be¬
fore he was able to calculate thefe tables; but his lofs
was in fome meafure fupplied by Mr Briggs, who applied
himfelf with zeal to this arduous talk, and publiffied in
1618 a table of the logarithms of all numbers from 1 to
1000. In 1624 he publiffied another table containing
the logarithms from 1000 to 20,000, and from 90,000
to 100,000. The defefts in Briggs’s tables were filled
up by his friends Gellibrand and Hadrian Vlacq, who
alfo publiffied new tables containing the logarithms of
fines, tangents, &c. for 90 degrees.
ofHaT"65 2^’ ^ur*n£ t^ie time when Napier and Briggs were
BornT-6o honour to their country by completing the fyllem
Died 1621! °f logarithms, algebra was making great progrefs in
the hands of our countryman Harriot. His Artis ana¬
lytic te Praxis, which appeared in 1620, contains along
with the difcoveries of its author, a complete view of
the Hate of algebra. He fimplified the nutation by
MAT ICS. 5
fubllituting fmall letters inftead of the capitals introdu¬
ced by Vieta ; and he was the firft who ffiowed that
every equation beyond the firft degree may be confider-
ed as produced by the multiplication of as many limple
equations as there are units in the exponent of the high-
eft power of the unknown quantity. From this he de¬
duced the relation which exifts between the roots of
any equation, and the coefficients of the terms of which
it confifts.
29. About the fame time, a foreign author named Fer- Fernel firft
nel, phyfician to King Henry II. of France, had theS:vesThe
merit of being the firft who gave the meafure of the J^earth0*
earth. By reckoning the number of turns made by a
coach wheel from Amiens to Paris, till the altitude of
the pole ftar was increafed one degree, he eftimated the
length of a degree of the meridian to be 56,746 toifes,
which is wonderfully near the truth. He alfo wrote a
work on mathematics, entitled De Proportionibus.—
About this time it was ffiown by Peter Metius, a German jy[ctiuS
mathematician, that if the diameter of a circle be 113, finds
its circumference will be 355. This refult, fo very near more cor-
the truth, and expreffed in fo few figures, has preferved J?um1"
the name of its author. dkmSer
30. The next author, whofe labours claim our atten-and circum-
tion, is the illuftrious Defcartes. We do not allude to ference of a
thofe wild and ingenious fpeculations by which this phi- circle*
lofopher endeavoured to explain the celeftial phenome- Difcoveries
na ; but to thefe great difcoveries with which he en- Defcartes
riched the kindred fciences of algebra and geometry.
He introduced the prefent method of marking the powers Died 165c.
of any quantity by numerical exponents. He firft ex¬
plained the ufe of negative roots in equations, and ffiow¬
ed that they are as real and ufeful as pofitive roots, the
only difference between them being founded on the dif¬
ferent manner in which the correfponding quantities are
confidered. He pointed out the method of finding the
number of pofitive and negative roots in any equation
where the roots are real; and developed the method
of indeterminates which Vieta had obfcurely hinted
at.
31. Though Regiomontanus, Tartalea, and Bombelli,
had refolved feveral geometrical problems by means of
alegbra, yet the general method of applying geometry
to algebra was firft given by Vieta. It is to Defcartes,IIe extends
however, that we are indebted for the beautiful and ex- tPe aPp^ca-
tenfive ufe which he made of his difcovery. His me• to "
thod of reprefenting the nature of curve lines by equa- geometry,
tions, and of arranging them in different orders accord¬
ing to the equations which diflinguiffied them, opened
a vaft. field of inquiry to fubfequent mathematicians;
and his methods of conftruifting curves of double cur¬
vature, and of drawing tangents to curve lines, have
contributed much to the progrefs of geometry. The '
in ‘crfe method of tangents, which it was referved for the
fluxionary calculus to bring to perfection, originated at
this time in a problem which Florimundus de Beaune
propofed to Defcartes. It was required to conftruft a A. D. 1647.
curve in which the ratio of the ordinate and fubtangent
ffiould be the fame as that of a given line to the por¬
tion of the ordinate included between the curve and a
line inclined at a given angle. The curve was con-
ftrufted by Defcartes, and feveral of its properties de¬
tected, but he was unable to accompliffi the complete A. D. 16;I
folution of the problem. Thefe difcoveries of Defcartes
were ftiidied and improved by his fucceffors, among
whom