Campanus's translation of Euclid was made from an Arabic manuscript ; but in 1505, Zamberti gave a transla tion from the original Greek. In the year 1518, the sphe rics 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 Mernmius. But although Zamberti and Meinmius might be good Greek scholars, they had little geometrical knowledge ; and hence their translations were in some measure imperfect. Com mandinus possessed both qualifications, and on that account succeeded better. He translated into Latin, and published in 1558, a part of the works of Archimedes, with a com mentary. The two books on floating bodies, of which the Greek text has never been found, were published by him in 1565. He gave, in the following year, the first four books of Apollonius'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 figures, by an Ara bian geometer : the original was furnished by John Dee, an English mathematician. But his last and most impor tant work was his translation of the mathematical collec tions 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 treasure of geometrical knowledge would still have been buried in the dust of libraries. Commandi nus 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 Mene laus from the Arabic, and two new books as a supplement. He prepared an edition, or rather imitation, of Archimedes, which was printed after his death ; and he treated of the conic sections, deducing them elegantly from the cone it self. He made the useful remark in dialling, that the sha dow of the top of a style describes a conic section on a plane.
Tartalea, one of the earliest cultivators of algebra, con tributed likewise to the revival of geometry. He made a translation of Euclid's Elements into Italian, which appear ed in 1547. Ile also gave a Latin translation of part of Archimedes in 1543 ; he demonstrated the rule for.finding the area of a triangle from its three sides ; but the rule it self is probably of great antiquity, as it occurs in the Gco desia of Hero the younger.
The very prolix commentary of Proclus on Euclid, was given in a Latin translation by two mathematicians, Napoli tain and Barozzi. And there were many other translators that would deserve notice in a history of geometry, if our limits would permit ; but we cannot find room to notice par ticularly all the cultivators of the science in the 16th cen tury. We shall therefore only mention a few ; as Clavius, whose translation and commentary on Euclid are still es teemed ; Benedictus, or Benedetto, mathematician to the Duke of Savoy, whose writings slim that he was well ac quainted with the ancient geometrical analysis ; Wolfius, who wished to demonstrate even the axioms of geometry ; and Ramus, the author of various esteemed works on the science.
The celebrated Vieta, who flourished in France towards the end of the 15th century, deserves particular notice. Ile was profoundly skilled in the ancient geometry, and he re stored the book of Tangencies of Apollonius, in his Apollo nius Gallus, an exquisite model of geometrical elegance. He was the first that carried 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 angu lar sections, one of the most elegant theories in The Low Countries produced several geometers of dis tinguished merit ; as Metius, who found a very convenient approximation to the ratio of the diameter of a circle to its circumference, viz. that of 113 to 355 ; and Adrianus Ro manus, a geometer much esteemed in his time. He car ried the approximation to the circumference'of the circles as far as 17 decimal figures ; and hence lie was the plague of all the pretenders to its quadrature ; for he was in every case able to shew, that the lines which they supposed equal to the circumference, were either greater than a polygon described about the circle, or else less than a polygon in scribed in it. In this way he refuted Joseph Scaliger, who imposed upon himself the task of squaring the circle as an amusement, 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 successful in sim plifying the number of cases.
Spain and Portugal can number only two geometers of note; the one was Nonius, or Nunez, who determined ele gantly the time of the shortest twilight, a problem 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.
At this period, England abounded in mathematicians. Robert Record, John Dee, Leonard and Thomas Digges, and H. Billingsley, all concurred in cultivating geometry. We are particularly indebted to Edward Wright for the in vention of his chart, which is improperly called Mercator'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, toast of John Werner of Nuremberg. He wrote on the conic sections; he attempted to restore Apollonius's treatise on the section of a ratio ; he also translated Euclid from Greek into German, and cultivated trigonometry. His Writings, however, have not been printed, Other German mathematicians did not cultivate so sublime a geometry. ltheticus extended the trigonometrical tables, and improv ed them by inserting the secants ; and Pitiscus still farther extended them in his Thesaurus Mathematieus sive Canon Sinuum, Ste. which contains the sines of every tenth second of the quadrant to 16 figures, and fur every second of the first and last degree to 26 figures, together 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 ac companied with much renown.