cro- Base 
ied 
Campanus, 
w~" of Theodosius from t 
GEOMETRY. 
voli, who, about the year 1120, translated the Spherics 
é Arabic into barbarous Latin. 
The thirteenth century was brilliant when compared 
with the ages that had gone before ; it was the twilight 
of that bright day which has enlightened Europe for 
upwards of 200 years. Among the mathematicians of 
this time may be reckoned John of Halifax, called also 
Sacro-Bosco, who wrote a treatise on the sphere, and 
Campanas of Navarre, the celebrated translator of Eue 
clid, and the author of a treatise on the quadrature of 
the circle ; in which he has supposed that the approxi- 
mate ratio found by Archimedes was quite exact; and 
proceeding on this, he has resolved some problems rela- 
ting to the circle: His paralogism. is excusable in consi- 
deration of the time in which he lived, The celebrated 
Albertus Magnus wrote on geometry in this century. 
It is instructive to reflect upon the principles in hu- 
man nature, by which, after ignorance has debased the 
mind, knowledge is again renovated. In the dark ages, 
when the true causes which bring about natural events 
Were unknown or but little understood, the principle 
in the mind, by which men are led to suppose co-existing 
events as somehow connected, made them conjecture 
_that the motions of the heavenly bodies, the most stri- 
king phenomena of nature, were closely connected with 
-the common events of life. In this way, probably, 
_astrology became a disease of the mind in the absence 
of genuine knowledge; but in pursuit of this delusion, 
it was necessary to cultivate astronomy, and this science 
again required the immediate aid of eometry. Thus 
we see, that from the very nature of the human under- 
standing, it has a tendency to emerge from ignorance, 
and that probably we are iadebtad. foe the restoration 
of the ancient astronomy and geometry to the vain spe- 
culations of judicial astrology. 
During the 14th century, England was fertile in ma- 
thematicians, They wrote treatises on arithmetic and 
Sony, but chiefly on astronomy. Their works, 
wever, have chiefly remained in the public libraries, 
The most remarkable was Richard Wallingfort, who 
raised himself from an obscure condition by his merit. 
The science of etry claims also the poet Chaucer 
as one of its cultivators. Even at this time, Britain 
ve indications of the approach of that brilliant wera of 
y van » Which will for ever render her illustrious 
as e be reg bs 
period now approac’ 
to recover more than tts origi 
pal promoters were then Purbach 
called also Regiomontanus. They 
trigonometry, and formed the resolution of travelling 
together into Ital 
Purbach dying, 
from the original. 
gave Latin versions of the nen of Menelaus, those 
er astronomical treatises : 
- besides, he corrected, by the Greek text, the ancient 
the Cylindries 
mathematicians, 
on certain books of Archimedes, which 
Eutocius had passed over: he defended Euclid’s defini- 
tion of proportionals against Campanus; and he refu- 
oe a pretended quadrature of the circle by Cardinal 
usa. 
Purbach rejected the ancient Sexagesimal division of 
the radius, and instead of it he supposed it to be divi- 
VOL. X. PART I. . 
He 
- 
; 193 
ded into 600,000, “Regiomontanus, again, improved on 
Purbach ; and, dividing the radius into 1,000,000 parts, 
he calculated new tables for every degree and minute 
of the quadrant, adding, for the first time, the tan- 
gents, it was Purbach that invented the geometrical 
square, and he appears to have been the first that ap- 
plied the plumb line to mark the divisions on instru- 
ments, 
Lucus Pacciolus, or De Burgo, must be reckoned one 
of the distinguished cultivators of geometry of this pe- 
riod, He revised Campanus’s translation of Euc id, 
but his labours did not appear until 1509. His work, 
Summa de Arithmetica Geometria, &c. 1494, contains a 
tolerable treatise on geometry. The progress which 
had now been made in the Greek tongue, and the in- 
vention of printing, contributed greatly to the dissemi. 
nation of geometrical knowledge.’ The Greek mathe 
maticians began now to be known in Europe; and Eu- 
clid was printed for the first time at Venice in 1482, in 
a folio volume, by Erhard Ratdolt, one of the first 
printers of the age: its title was, Preclarissimus liber 
Elementorum Euclidis perspicacissimi in artem geometrie 
incipit quam felicissime. And at the end we read, Opus 
Elementorum Euclidis Megarensis in geometricam artem ; 
in id quogue Campani persptcassimi commentationes. 
Erhardus Ratdolt, “Augusiensis impressor Solertissimus, 
Veneliis impressit, anno salutis MCCCCLXXKXII. Oct. 
cal. Junii. Lector vale. On the back of the title-page, 
there is a dedication to the reignin Doge. 
Campanus’s translation of Euclid was made from an 
Arabic manuscript ;_ but in 1505, Zamberti gave a trans- 
lation from the original Greek. In the year 1518, the 
spheries of Theodosius appeared for the first time; and 
in 1537, there came out a translation of the first four 
books (the only ones then known), of Apollonius by 
Memmius. But although Zamberti and Memmius might 
be good Greek scholars, they had little geometrical 
knowledge ; and hence their translations were in some 
measure imperfect. Commandinus possessed both qua- 
lifications, and on that account succeeded better. He 
translated into Latin, and published in 1558, a part of 
the works of Archimedes, with a commentary. The 
two books on floating bodies, of which the Greek text 
has never been found, were published by him in 1565. 
ve, inthe following year, the first four books of 
lonius’s conics, with the commentary of Eutocius, 
and the lemmas of Pappus. His Latin translation. of 
Euclid appeared in 1572. Geometry is also indebted 
to him for a treatise on Geodesia, or the divisions of fi- 
gures, by an Arabian geometer: the original was fur- 
nished by John Dee, an English mathematician, But 
his last and most important work was his translation of 
the mathematical eolleckone of Pappus, the only one 
that has yet appeared. It is probable that, had it not 
been for his zeal in the cause of mathematics, this trea- 
sure of geometrical knowledge would still have been 
buried in the dust of libraries. Commandinus died in 
1575, and his Pappus was printed after his death in 
1588. ; 
Maurolycus of Messina distinguished himself both 
_ by his editions of the ancients and his original works. 
In 1558, he published a new translation of the spherics 
of Theodosius from the Greek ; to this he joined the 
spherics of Menelaus from the Arabic, and two new 
books as a supplement. He prepared an edition, or ra- 
ther imitation of Archimedes, which was printed’ after 
his death ; and he treated of the conic sections, dedu- 
cing them elegantly from the cone itself. He made the 
2B 
History. 
— a 
De Burgo. 
1480. 
Commandi« 
nus. Born 
1509. Died 
1575. 
Mauroly- 
eus. Born 
1495. Died 
1575, 
