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books. His successor, Marinus, was the author of a Preface to Euclid's Data, which is generally prefixed to the head of that work. Hero also has given, in his Geodesia, the method of finding the area of a triangle by means of its three sides, without knowing the perpendicular.

Having thus far exhibited to the reader those votaries of science who flourished in the first period of her history, with their discoveries and improvements, let us turn back to Euclid, the celebrated author of the Elements of Geometry, which bear his name. The origin, and country of Euclid, are not fully known; although he is generally supposed to have lived in the time of Ptolemy Lagus, about 270 years before the Christian era; and it is also generally believed that he studied at Athens, under the disciples of Plato.

No book of science ever met with success equal to that of Euclid's Elements. They have been taught for several centuries in every mathematical school of eminence, and translated and commented upon in all languages.

With regard to the composition of the work, it is evident from the authority of Proclus, and other ancient writers, that many of the propositions contained in it were known at a very early period, and that elementary treatises on geometry had been composed by Hippocrates of Chios, Eudoxus, and many others, which doubtless rendered him considerable services in the formation of his Elements. As a proof of the superior excellence of this work, Euclid has deduced from a few first principles or

axioms, a complete series of the most useful propositions in the science. His demonstrations are so very nervous and elegant, as not to be equalled by any geometrical writer, ancient or modern; and his method is such that nothing is taken as true unless demonstrated; and nothing is demonstrated, but from what went before. In consequence of this rigorous system of demonstration, it is reported that king Ptolemy, once asking Euclid whether there was no shorter way of arriving at geometry than by these his Elements, is said to have answered, There is no other way or royal road to Geometry.

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Of the numberless editions of this valuable work, the following have met with the most considerable encouragement for their accuracy and superior excellence.

Campanus translated the whole fifteen books of the Elements into Latin, from the Arabic, in 1482.

Zambertus translated from Greek into Latin, the fifteen books and Data. This edition was edited at Paris in the year 1516; also at Basil in 1537, and 1546. The Data are only in the two last editions.

Candalla edited a Latin translation of the fifteen books in 1566.

Commandine, one of the best geometers of his age, translated into Latin the fifteen books from the Greek text of the Basil edition.

The Greek text of the Data of Euclid, with the Latin translation of Hardiæus, was edited in 1625.

A superb edition of all the works of Euclid, was edited in 1703, by Dr. David Gregory, in Greek and Latin, under the title of Euclidis quæ supersunt omnia.

Peyrard edited at Paris the fifteen books and Data in 1818, which is esteemed the best edition as to correctness and purity of text.

In English, we have Billingsley's, Barrow's, Keill's, Stone's, Simson's, &c. editions, which possess great merit, and which do honour to the talents of their respective editors.

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EUCLID'S ELEMENTS.

BOOK I.

DEFINITIONS.

1. A POINT is that which has no magnitude, or is no part of any thing.

2. A line is length without breadth.

3. The extremities of a line are points.

4. A right line is that which lies evenly between its extreme points.

5. A superficies is that which has only length and breadth.

6. The extremities of a superficies are lines.

7. A plane superficies is that which lies evenly between its lines.

8. A plane angle is the mutual inclination of two lines to one another in the same plane, so touching each other as not both to lie in the same right line.

9. When the lines containing the said angle are right lines, it is called a rectilineal angle.

10. When a right line standing on another right line, makes the adjacent angles equal to one another, each of the equal angles is a right angle, and the right line standing on the other is called a perpendicular.

11. An obtuse angle is that which is greater than a right angle.

12. An acute angle is that which is less than a right angle.

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