GEOMETRY

, the science or doctrine of local extension, as of lines, surfaces, and solids, with that of ratios, &c.

The name Geometry literally signifies measuring of the earth, as it was the necessity of measuring the land that first gave occasion to contemplate the principles and rules of this art; which has since been extended to numberless other speculations; insomuch that together with arithmetic, Geometry forms now the chief foundation of all the mathematics.

Herodotus (lib. 2), Diodorus (lib 1), Strabo (lib. 17), and Proclus, ascribe the invention of Geometry to the Egyptians, and assert that the annual inundations of the Nile gave occasion to it; for those waters bearing away the bounds and land-marks of estates and farms, covering the face of the ground uniformly with mud, the people, say they, were obliged every year to distinguish and lay out their lands by the consideration of their figure and quantity; and thus by experience and habit they formed a method or art, which was the origin of Geometry. A farther contemplation of the draughts of figures of fields thus laid down, and plotted in proportion, might naturally lead them to the discovery of some of their excellent and wonderful properties; which speculation continually improving, the art continually gained ground, and made advances more and more towards perfection.

Josephus however seems to ascribe the invention to the Hebrews: and others of the ancients make Mercury the inventor. Polyd. Virgil, de Invent. Rer. lib. 1, cap. 18.

From Egypt, this science passed into Greece, being carried thither by Thales; where it was much cultivated and improved by himself, as also by Pythagoras, Anaxagoras of Clazomene, Hippocrates of Chios, and Plato, who testified his conviction of the necessity and importance of Geometry to the successful study of philosophy, by this inscription over the door of his academy, Let no one ignorant of Geometry enter here. Plato thought the word Geometry too mean a name for this science, and substituted instead of it the more extensive name of Mensuration; and after him others gave it the title of Pantometry. But even these are now become too scanty in their import, fully to comprehend its extent; for it not only inquires into, and demonstrates the quantities of magnitudes, but also their qualities, as the species, figures, ratios, positions, transformations, descriptions, divisions, the finding of their centres, diameters, tangents, asymptotes, curvature, &c. Some again define it as the science of inquiring, inventing, and demonstrating all the affections of magnitude. And Proclus calls it the knowledge of magnitudes and figures, with their limitations; as also of their ratios, affections, positions, and motions of every kind.

About 50 years after Plato, lived Euclid, who collected together all those theorems which had been invented by his predecessors in Egypt and Greece, and digested them into 15 books, called the Elements of Geometry; demonstrating and arranging the whole in a very accurate and perfect manner. The next to Euclid, of those ancient writers whose works are extant, is Apollonius Pergæus, who flourished in the time of Ptolomy Euergetes, about 230 years before Christ, and about 100 years after Euclid. He was author of the first and principal work on Conic Sections; on account of which, and his other accurate and ingenious geometrical writings, he acquired from his patron the emphatical appellation of The Great Geometrician. Contemporary with Apollonius, or perhaps a few years before him, flourished Archimedes, celebrated for his mechanical inventions at the siege of Syracuse, and not less so for his very many ingenious Geometrical compositions.

We can only mention Eudoxus of Cnidus, Archytas of Tarentum, Philolaus, Eratosthenes, Aristarchus of Samos, Dinostratus, the inventor of the quadratrix, Menechmus, his brother and the disciple of Plato, the two Aristeus's, Conon, Thracidius, Nicoteles, Leon, Theudius, Hermotimus, Hero, and Nicomedes the in-| ventor of the conchoid: besides whom, there are many other ancient geometricians, to whom this science has been indebted.

The Greeks continued their attention to it, even after they were subdued by the Romans. Whereas the Romans themselves were so little acquainted with it, even in the most flourishing time of their republic, that Tacitus informs us they gave the name of mathematicians to those who pursued the chimeras of divination and judicial astrology. Nor does it appear they were more disposed to cultivate Geometry during the decline, and after the fall of the Roman empire. But the case was different with the Greeks; among whom are found many excellent Geometricians since the commencement of the Christian era, and after the translation of the Roman empire. Ptolomy lived under Marcus Aurelius; and we have still extant the works of Pappus of Alexandria, who lived in the time of Theodosius; the commentary of Eutocius, the Ascalonite, who lived about the year of Christ 540, on Archimedes's mensuration of the circle; and the commentary on Euclid, by Proclus, who lived under the empire of Anastasius.

The consequent inundation of ignorance and barbarism was unfavourable to Geometry, as well as to the other sciences; and the few who applied themselves to this science, were calumniated as magicians. However, in those times of European darkness, the Arabians were distinguished as the guardians and promoters of science; and from the 9th to the 14th century, they produced many astronomers, geometricians, geographers, &c; from whom the mathematical sciences were again received into Spain, Italy, and the rest of Europe, somewhat before the year 1400. Some of the carliest writers after this period, are Leonardus Pisanus, Lucas Paciolus or De Burgo, and others between 1400 and 1500. And after this appeared many editions of Euclid, or commentaries upon him: thus, Orontius Finæus, in 1530, published a commentary on the first 6 books; as did James Peletarius, in 1557; and about the same time Nicholas Tartaglia published a commentary on the whole 15 books. There have been also the editions, or commentaries, of Commandine, Clavius, Billingsly, Scheubelius, Herlinus, Dasypodius, Ramus, Herigon, Stevinus, Saville, Barrow, Taquet, Dechales, Furnier, Scarborough, Keill, Stone, and many others; but the completest edition of all the works of Euclid, is that of Dr. Gregory, printed at Oxford 1703, in Greek and Latin: the edition of Euclid, by Dr. Robert Simson of Glasgow, containing the first 6 books, with the 11th and 12th, is much esteemed for its correctness. The principal other elementary writers, besides the editors of Euclid, are Borelli, Pardies, Marchetti, Wolfius, Simpson, &c. And among those who have gone beyond Euclid in the nature of the Elementary parts of Geometry, may be chiefly reckoned, Apollonius, in his Conics, his Loci Plani, De Sectione Determinata, his Tangencies, Inclinations, Section of a Ratio, Section of a Space, &c; Archimedes, in his treatises of the Sphere and Cylinder, the Dimension of the Circle, of Conoids and Spheroids, of Spirals, and the Quadrature of the Parabola; Theodosius, in his Spherics; Serenius, in his Sections of the Cone and Cylinder; Kepler's Nova Stereometria; Cavalerius's Geometria Indivisibilium; Torricelli's Opera Geometrica; Viviani, in his Divinationes Geometricæ, Exercitatio Mathematica, De Locis Solidis, De Maximis & Minimis, &c; Vieta, in his Effectio Geometrica, Supplement. Geometriæ, Sectiones Angulares, Responsum ad Problema, Apollonius Gallus, &c; Gregory St. Vincent's Quadratura Circuli; Fermat's Varia Opera Mathematica; Dr. Barrow's Lectiones Geometricæ; Bulliald de Lineis Spiralibus; Cavalerius; Schooten and Gregory's Exercitationes Geometricæ, and Gregory's Pars Universalis, &c; De Billy's treatise De Proportione Harmonica; La Lovera's Geometria veterum promota; Slusius's Mesolabium, Problemata Solida, &c; Wallis, in his treatises De Cycloide, Cissoide, &c; De Proportionibus, De Sectionibus Conicis, Arithmetica Infinitorum, De Centro Gravitatis, De Sectionibus Angularibus, De Angulo Contactus, Cuno-Cuneus, &c, &c; Hugo De Omerique, in his Analysis Geometrica; Pascal on the Cycloid; Step. Angeli's Problemata Geometrica; Alex. Anderson's Suppl. Apollonii Redivivi, Variorum Problematum Practice, &c; Baronius's Geomet. Prob. &c; Guido Grandi Geometr. Demonstr. &c; Ghetaldi Apollonius Redivivus, &c; Ludolph van Colen or a Ceulen, de Circulo et Adscriptis, &c; Snell's Apollonius Batavus, Cyclometricus, &c; Herb<*>rstein's Diotome Circulorum; Palma's Exercit. in Geometriam; Guldini Centro-Baryca<*> with several others equally eminent, of more modern date, as Dr. Rob. Simson, Dr. Mat. Stewart, Mr. Tho. Simpson, &c. Since the introduction of the new Geometry, or the Geometry of Curve Lines, as expressed by algebraical equations, in this part of Geometry, the following names, among many others, are more especially to be respected, viz, Des Cartes, Schooten, Newton, Maclaurin, Brackenridge, Cramer, Cotes, Waring, &c, &c.

As to the subject of Practical Geometry, the chief writers are Beyer, Kepler, Ramus, Clavius, Mallet, Tacquet, Ozanam, Wolfius, Gregory, with innumerable others.

Geometry is distinguished into Theoretical or Speculative, and Practical.

Theoretical or Speculative Geometry, treats of the various properties and relations in magnitudes, demonstrating the theorems, &c. And

Practical Geometry, is that which applies those speculations and theorems to particular uses in the solution of problems, and in the measurements in the ordinary concerns of life.

Speculative Geometry again may be divided into Elementary and Sublime.

Elementary or Common Geometry, is that which is employed in the consideration of right lines and plane surfaces, with the solids generated from them. And the

Higher or Sublime Geometry, is that which is employed in the consideration of curve lines, conic sections, and the bodies formed of them. This part has been chiefly cultivated by the moderns, by help of the improved state of Algebra, and the modern analysis or Fluxions.