194 
History. useful remark in dialling, that the shadow. of the top of 
—y~" a style describes a conic section on a plane. 
Wencalen Tartalea, one of the earliest cultivators of algebra, 
Born 1479. contributed likewise to the revival of geometry. He 
Died 1557. made a translation of Euclid’s Elements into Italian, 
which appeared in 1547. He also gave a Latin trans- 
lation of part of Archimedes in 1543 ; he demonstrated 
the rule for finding the area of a triangle from its three 
sides ; but the rule itself is probably of great antiquity, 
as it occurs in the Geodesia of Hero the younger. 
The very prolix commentary of Proclus on Euclid, 
was given in a Latin translation by two mathemati- 
cians, Napolitain and Barozzi. And there were many 
other translators that would deserve notice in a history 
of geometry, if our limits would permit; but we can- 
not find room to notice particularly all the cultivators 
-of the science in the 16th century. We shall therefore 
€lavius.  -only mention a few ; as Clavius, whose translation and 
Born 1537. gommentary on Euclid are still esteemed ; Benedictus, 
Died 1612. (Benedetto, mathematician to the Duke of Savoy, 
whose writings shew that he was well acquainted with 
the ancient geometrical analysis ; Wolfius, who wish- 
ed to demonstrate even the ‘axioms of yeometry ; and 
meaty ;, Ramus, the author of ‘various esteemed works on the 
Died 1572, "Science. t : : 
eae The celebrated Vieta, who flourished in France to- 
1e' 
hom 1540 wards the end of the 15th century, deserves particular 
Died 1603," 20tice. He was profoundly skilled in the ancient geo- 
metry, and he’ restored the book of Tangencies of 
Apollonius, in his Apollonius Gallus, an exquisite mo- 
del of geometrical elegance. H< was the first that car- 
ried the approximate value ‘of the ratio of the diameter 
of'a circle to its circumference as far as. eleven figures ; 
and ‘to him we owe the doctrine of angular sections, 
one of the most elegant theories in geometry. 
The Low Countries produced several geometers of 
Metius, distinguished merit ; as Metius, who found a very con- 
Died 1636. venient ‘approximation to the ratio of the diameter of a 
circle to its circumference, viz. that of 113 to 355 ; and 
Romanus. Adridnus Romanus, a geometer much esteemed in his 
Died 1625. time, He carried the approximation to the circumfer- 
ence of the circles as far as 17 decimal figures; and 
hence he was the plague of'all the pretenders to its 
a a rae for he was in every case able to shew, that 
e lines which they supposed equal to the circumfe- 
rence, were either greater than a polygon described about 
the circle, or else less than a polygon inscribed in it. 
In this way he refuted Joseph Scaliger, who imposed 
~upon himself the task of squaring the circle as an amuse- 
‘ment, just to shew his superiority to the plodding ma- 
thematicians, who had long sought it in vain. He 
wrote a treatise on Trigonometry, and was very suc- 
cessful in simplifying the number of cases. 
Spain and Portugal can number only two geometers 
Nonius. of note; the one was Nonius, or Nunez, who deter- 
Born 1497. ‘rained elegantly the time of the shortest twilight, ‘a pro- 
Died 1577. ‘blem which seems for a long'time to have puzzled James 
Bernoulli. The other was John of Royas, a Castilian, 
the inventor of a projection of the sphere. 
Record, At this period, England abounded in mathematicians. 
Died 1558. Robert Record, John Dee, Leonard and Thomas Digges, 
Dee. Died @2d H. Billingsley, all concurred in cultivati 
‘geome- 
1608. try. We are particularly indebted to Edwa Wright 
Wright, for the invention of his chart, which is improperly call- 
Died 1615. €d Merecator’s. His book on the correction of certain 
errors in navigation, indicates «a geometry beyond that 
of his time, 
Germany then produced but few geometers ; it might, 
however, boast. of John Werner of Nuremberg. He 
: ’ 
GEOMETRY. 
_ Scottish geometers, 
wrote on the conic sections; he attempted to restore History 
Apollonius’s treatise on the section of a ratio; he also ; 
translated Euclid from Greek into German, and culti- oe 146 
vated trigonometry. His writings, however, have not pica 153 
been printed. Other German mathematicians did not ; 
cultivate so sublime a geometry. Rheticus extended Rheticu 
the trigonometrical tables, and improved them by in- Bom 15 
serting the secants ; and Pitiscus still farther extended 13 
them in his Thesaurus Mathematicus sive Canon Sinuum, Pitiscus. 
&c. which contains the sines of every tenth second of 16 
the quadrant to 16 figures, and for every second of the 
first and last degree to 26 figures, ther with the first, 
‘second, and in some cases the third differences. This 
is one of the most remarkable monuments of human 
patience, and is so much the more meritorious, that it 
was not accompanied with much renown, 
We now enter upon the 17th century, the most fer- 
tile of any in atical discoveries: in fact, the 
progress since made in the science is little more than 
their expansion ; and whatever perfection it may attain 
in future ages, a t share the glory will belong 
to the period at which we are now a R mt 
» One of the earliest geometers of the 17th century was L, Valer 
Lucas Valerius, an Italian, and Professor of Mathema- 1604, _ 
tics at Rome. ‘He determined the centre of gravity in 
complete conoids and ‘spheroids, as well as in ae 
ments cut off by planes parallel to the base. Ar 
medes had resolved this lem only in the case of 
the parabolic conoid ; ‘Commandinus had extended 
the subject a little farther, to the easiest cases; but 
Valerius went beyond them both. 
Marinus Ghetaldus, a native of Ragusa, was ‘well Ghetaldus 
acquainted with the ancient geometry. Guided by the Pied 160! 
indications of Pappus, he attempted to restore ‘the lost be 
book of Apollonius on IJnclinations, in a work called 
Apollonius Redivivus. He also wrote a supplement to 
the Apollonius ‘Gallus of Vieta. He died on a mission 
to Turkey in 1609. ? J weary. 
Alexander Anderson was one of the earliest of the ang 
He a sto have been a friend 
or scholar of Vieta, some of whose ’ s works 
he published. .He was well acquainted with the geo- 
metrical analysis; and of this he has given proof in his 
Supplementum Apollonii Redivivi, where he-endeavours 
to supply what Ghetaldus has left incomplete ‘in ‘his 
work. See ANDERSON. see “- 
The Low Countries produced ‘in this period ‘several Ceulen, 
mathematicians, whose labours were conducive to the 1619. 
progress of geometry. Ludolph Van Ceulen claims at- 
tention, on account of the immensely long calculation = __ 
by which he determined that the diameter of a circle 
being supposed 1, the circumference will be ‘between 
the number 
8.14159,26535,89793,23846,26433,83279,50288, 
and the same number increased by unity in the last 
figure. It must be acknowledged ‘that there was more 
patience than genius displayed in ‘this effort ; for he 
proceeded simply after the manner of Archimedes, in- 
scribing polygons in a:circle, and describing others of 
an equal number of sides about ‘it, until he found = 
inscribed and circumseribin gon ‘to ‘agree in‘ 
figures. After the example of lf desired 
that this, his greatest discovery, should be inscribed on 
his tomb. Geometry, however, derived more real ser- 
vice from his other labours. feat ; 
Willebrod Snellius was another of the Dutch ma- Snellius 
thematicians: At the age of seventeen, he undertook to Born lé 
restore Apollonius’s book of Determinate Sections, and Died! 
