A similar Rocess enables us to discover the demonstrations of propositions, supposed to be true, or, if not true, to discover that they are false.
This method, to the consideration of which we shall again have an opportunity of re turning, was perhaps the most valuable part of the ancient mathematics, inasmuch as a method of discovering truth is more valuable than the truths it has already discovered. Unfortunately, however, the fragments containing this precious remnant had suffered more from the injuries of time than almost any other.
In the fifteenth century, Regiomontanus, already mentioned, is the mathematician who holds the highest rank. To him we owe many translations and commentaries, together with several original and valuable works of his own. Trigonometry, which had never been known to the Greeks as a separate science, and which took that form in Arabia, advanced, in the hands of Regiomontanus, to a great degree of perfection, and approached very near to the condition which it has attained at the present day. He also introduced the use of decimal fractions into arithmetic, and thereby gave to that scale its full extent, and to numerical computation the utmost degree of simplicity and enlargement which it seems capable of attaining.
This eminent man was cut off in the prime of life ; and his untimely death, says Mr Smith, amidst innumerable projects for the advancement of science, is even at this day a matter of regret.' He was buried in the Pantheon at Rome ; and the honours paid to him at his death prove that science had now become a distinction which the great were disposed to recognise.
Werner, who lived in the end of this century, is the first among the moderns who ap pears to have been acquainted with the geometrical analysis. His writings are very rare, and I have never had an opportunity of examining them. What I here assert is on the authority of Montucla, whose judgment in this matter may be safely relied on, as he has shown, by many instances, that he was well acquainted with the nature of the ana lysis referred to. It is not a little remarkable that Werner should have understood this subject, when we find many eminent mathematicians, long after his time, entirely unac quainted with it, and continually expressing their astonishment how the ancient geometers found out those simple and elegant constructions and demonstrations, of which they have given so many examples. In the days of Werner, there was no ancient book known ex
cept the Data of Euclid, from which any information concerning the geometrical analysis could be collected ; and it is highly to his credit, that, without any other help, he should have come to the knowledge of a method not a little recondite in its principles, and among the finest inventions either of ancient or of modern science. Werner resolved, by means of it, Archimedes's problem of cutting a sphere into two segments, having a given ratio to one another. He proposed also to translate, from the Arabic, the work of Apollonius, entitled Sec& Rationis, rightly judging it to be an elementary work in that analysis, and to come next after the Data of Euclid.* Benedetto, an Italian mathematician, appears also to have been very early acquainted with the principles of the same ingenious method, as he published a book on the geome trical analysis at Turin in 1585.
Maurolycus of Messina flourished in the middle of the sixteenth century, and is justly regarded as the first geometer of that age. Beside furnishing many valuable translations and commentaries, he wrote a treatise on the conic sections, which is highly esteemed. He endeavoured also to'restore the fifth book of the conics of Apollonius, in which that geo meter treated of the maxima and minima of the conic sections. His writings all indicate a man of clear conceptions, and of a strong understanding ; 'though he is taxed with hav ing dealt in astrological prediction.
In the early part of the seventeenth century, Cavalleri was particularly distinguished, and made an advance in the higher geometry, which occupies the middle place between the discoveries of Archimedes and those of Newton.