GEOMANTIA, a kind of divination, performed by means of a number of little points or dots, made on paper at random, and considering the various lines and figures which those points present, and thence forming a pretended judgment of futurity, and deciding any question proposed. The word is formed from the Greek γῆ, or γῆ, terra, earth, and ἀνάγνωσις, divination; the ancient custom being to cast little pebbles on the ground, and thence to form conjectures, instead of the points afterwards made use of. Polydore Virgil defines geomancy to be a kind of divination performed by means of clefts or chinks made in the ground, and of which he conceives the Persian Magi to have been the inventors. The properties of bodies may be resolved into two classes; one comprehending those which belong to all bodies whatsoever, and another, such as belong only to particular bodies. Amongst the properties of the first class we may consider extension, magnitude, figure, divisibility, impenetrability, inertia, weight, mobility, &c. Some of those of the second are solidity, liquidity, transparency, and the like. Of these properties, extension, magnitude, figure, and divisibility, are the subject of Geometry. The discussion of other properties of body belongs to Physics, called also Natural Philosophy.

The objects of geometry having been continually presented to the human mind, and being of such a nature as could at all times be perfectly comprehended, it may be supposed that the science would in some form or other exist from the very beginning of society. Indeed there is a natural geometry which all possess, and which must have been employed in the divisions of property and the erection of dwellings in every age. The name of the science indeed indicates its origin, for it is derived from γεωμετρία, the science of land-measuring.

According to the testimony of Herodotus, the science was first cultivated in Egypt. He had been informed at Memphis and Thebes, that Sesostris the king had divided the lands amongst his subjects, giving to each an allotment, for which an annual tribute was to be paid. The overflowing of the Nile, however, disturbed the landmarks, and rendered it necessary to re-adjust them by measurement; and hence the origin of geometry, which in the course of time passed from Egypt into Greece. There are two things to be here noticed; the assertion of a verification depending on geometry, and the particular opinion of Herodotus as to the origin of the science. If, as some chronologers have supposed, Sesostris be the same as the King Sessac, or Shishac, who made war on Rehoboam, the son of Solomon, it would follow from Herodotus, that geometry had its origin not more than about a thousand years before the Christian era. It might, however, be earlier; for the operation of measuring the land indicated a science which had made some progress. Aristotle has attributed the invention of geometry to the Egyptian priests, who, living secluded from the world, had leisure for study. But however this might be, all ancient writers are agreed in giving the Egyptians the credit of having been the earliest cultivators of geometry.

The philosopher Thales, who lived 640 years before Christ, brought the sciences, and particularly mathematics, into Greece from Egypt, whither he had gone in quest of knowledge at an advanced period of life. Diogenes Laertius relates that he there measured the height of the obelisks by means of their shadows; and Plutarch says, that this display of skill in geometry astonished the king Amasis. This shows that the Egyptians had not then advanced far in geometrical knowledge. Proclus also has recorded that Thales, by geometrical principles, determined the distance of vessels remote from the shore. On his return to Greece he founded the Ionian school, so called from Ionia, his native country. His celebrity for learning excited the attention of his countrymen, and drew to him disciples.

There appears to have been some slight traces of geometry in Greece at a still earlier period. We allude to the geometrical construction of a triangle by Euphorbus of Phrygia, and the discovery of some properties of figures. These, however, were probably just the natural geometry common to all ages. The rule and compasses, most important instruments, have been referred to the ages of fable, the honour of the invention of the latter having been ascribed to the nephew of Dedalus, and that of the square and level to Theodorus of Samos, one of the architects of the temple of Ephesus. But there are no certain indications of geometry as a science before Thales. He laid its foundation in Greece, and infused into his countrymen a taste for the science. He is said to have applied a circle to the measurement of angles, and to have discovered various properties of triangles, by comparing them one with another. In particular, he discovered the important proposition that all angles in a semicircle are right angles, a discovery which greatly delighted him, and for which he expressed his gratitude to the Muses by a sacrifice. Proclus has recorded that he made many discoveries in the science, which, however, have gone to the common stock of geometrical knowledge that had accumulated before Euclid collected it into a system, and the authors of which are unknown. It is probable that the greater part of the disciples of Thales were geometers, but few of their names have descended to us. Ameristus and Anaximander are the only ones now known. Of the former we only know that he was a skilful geometer. The latter wrote a kind of elementary treatise or introduction to geometry, the earliest work of the kind that is known to have existed.

Thales was succeeded in his school by Anaximander, who is said to have invented the sphere, the gnomon, and geographical charts. These required a considerable knowledge of geometrical science. It has been said that he attempted the grand practical geometrical problem of measuring the magnitude of the earth. This probably has originated in his being the inventor of representations of its surface. The honour of being the first to measure the earth seems to belong to another. Anaximander was followed by Anaximenes, born at Miletus 540 years B.C. He is said to have invented sun-dials, and thus must have known both astronomy and geometry; indeed a desire to understand the former would prove a strong stimulus to the study of the latter. He had as his disciple and successor Anaxagoras, who being persecuted for his opinions, employed himself in prison in trying to square the circle. This is the first notice on record of an attempt to resolve what appears now to be an insoluble problem, on which no modern geometer will spend his time. It might, however, have a very different aspect to the ancient mathematicians.

Pythagoras, born about 580 years B.C., was one of the greatest men of antiquity. According to some he was a Tuscan, whilst others say he was a Tyrian; but his name is better known than his origin. At the age of eighteen he became the disciple of Thales, and imbibed deeply his opinion of the importance of temperance and economy of time as necessary to success in the study of philosophy. In the infancy of science knowledge was only to be acquired by travel, the men of different countries standing in the place of our modern books as the depositories of learning. Pythagoras visited Phoenicia, Chaldea, and India. In the annals of this last country the remembrance of the philosophical traveller is still preserved. He afterwards went into Egypt, where he is said to have remained twenty-two years, holding intercourse with the priests. The long duration of his abode is not probable, as the knowledge he seems to have acquired did not correspond to so protracted a residence. Whilst Egypt, he is said to have consulted the columns of Sothis, which celebrated person had inscribed the principles of geometry. He returned to Samos, his adopted country, and there opened his school; but not finding auditors, he removed to Italy, and took up his abode in Cortona, a town in the territory of Tarentum, where he found many disciples, and acquired a great reputation. He is said to have discovered the important theorem in geometry, that a right-angled triangle the squares on the sides containing the right angle are together equal to the square on the side opposite to it; and on this account he is said to have sacrificed an hundred oxen in gratitude to the Muses. The sacrifice is probably a fable. The shedding of blood was contrary to his moral principles, and so many oxen were not likely to be at the disposal of a philosopher. In his school, geometry made great progress, and was augmented by several new theories, such as that of the incommensurability of certain lines, in particular that of the side of a square to its diagonal. The theory of the regular solids had its origin also in the Pythagorean school, a doctrine which requires an extensive knowledge of geometry. This subject, now neglected amidst the riches of modern mathematics, because of its small utility, was in the beginning of great importance, on account of other discoveries which must have been incidentally made in its prosecution. Diogenes Laertius has attributed to Pythagoras the discovery that all figures having the same boundary, the circle among plane figures, and the sphere among solids, are the most capacious. If this was so (a thing doubtful), he is the first on record who considered isoperimetrical problems.

The Pythagorean school sent forth many philosophers and mathematicians; amongst the latter, Philolaus and Archytas held a distinguished rank. Of Philolaus we know little. He held to its full extent the Pythagorean doctrine of the motion of the earth round the sun as a centre, and was the first to unveil it. He composed a work on mechanics, and thus has a claim to be associated with Euclid and Archytas as inventors of that part of mathematics. He had a tragic end, having been massacred by the people of the small republic of which he had been a citizen. We know more of Archytas. He was the author of a solution of the problem of two mean proportions. He was also one of the first who made use of the geometrical analysis, which, according to Proclus, he had learned from Plato, and by its assistance he made many discoveries in geometry. We pass over his mechanical inventions, one of which, the artificial pigeon, which winged its way through the air, certainly savours of the marvellous. Archytas is said to have been blamed by Plato for applying geometry to mechanics; but if this was so, it might be for the manner of the application. His solution of the problem of finding two mean proportions gives support to this conjecture; for although ingenious, it has the defect of requiring a motion which cannot be executed.

Democritus of Abdera cultivated geometry extensively. It has been conjectured that he was one of the first who treated of the contact of circles and spheres, and of irrational lines and solids.

Hippocrates, who lived about the year 380 B.C., was originally a merchant; but having no turn for commerce, he did not prosper. To renovate his affairs, he went to Athens, and there for the first time became acquainted with geometry, which seemed to suit his particular turn of mind. In the prosecution of this study he discovered that the curvilinear space, called from its form a lune, comprehended half the circumference of one circle and the fourth of the circumference of another (their concavities being turned the same way), was equal to a rectilineal space, viz. the area of a right-angled triangle, whose hypotenuse was the diameter of the circle, and its sides each equal to the radius of the quadrantial arc. This could not be considered as a true quadrature of a curvilinear space; it was merely a geometrical juggle, by which a common space being taken from two equal curvilinear figures, one of the remainders was by a sort of chance a rectilineal figure. The first true quadrature of a curvilinear space is due to Archimedes. Hippocrates attempted the quadrature of the circle; but, if his mode of reasoning has been truly handed down to us, he committed a great oversight. This is the first recorded paralogism in geometry. He was more successful in treating of the duplication of the cube, which he showed to depend on the finding of two mean proportionals between two given lines. He was also the first who composed elements of geometry, but these have been lost. He appears to have retained the mercantile spirit, insomuch that he accepted money for teaching geometry. On this account he was expelled the Pythagorean school; a measure somewhat hard, considering his reduced circumstances.

Two geometers, Bryson and Antiphon, appear to have lived about the time of Hippocrates, and a little before Aristotle. They are only known by some animadversions of the latter on their attempts to square the circle. Before this time geometers knew that the area of a circle was equal to a triangle whose base is equal to its circumference, and altitude equal to its radius. This truth could not escape the observation of the early geometers. Bryson erroneously believed that, by a geometrical construction, he could find the circumference, in which case the quadrature would have been easy. Antiphon, it appears, proceeded by increasing continually the number of sides of an inscribed regular polygon, and considered the circle to be equal to the ultimate result; a process perfectly correct, and the same in effect as that by which Archimedes found the area of the parabola.

From this brief view of the progress of geometry during the first two centuries after its introduction into Greece, we pass to the school of Plato, in which geometry completely changed its character, and advanced with increased vigour and more rapid strides. Hitherto its subjects had been of the most elementary nature; now, however, the views of geometers became more enlarged, and a new era in the science commenced. Although the disciple and successor of Socrates, who had no taste either for mathematical or physical science, Plato held them in the highest estimation; and, imitating the earlier sages of Greece, he undertook voyages and journeys to improve his mathematical knowledge. He visited Egypt to converse with the priests, and Italy to consult with the celebrated Pythagoreans Philolaus, Timaeus of Locris, and Archytas. With the last of these he contracted a particular friendship. He went also to Cyrene to hear the discourses of the mathematician Theodorus; and on his return to Greece, when he had founded his school, he made the mathematics, particularly geometry, the basis of his instructions; and his disciples, encouraged by his example and his exhortations, applied with ardour to the study of that science. He never allowed a day to pass without making his disciples acquainted with some new truth. He placed over his school this inscription, *Let no one ignorant of geometry enter here.* He held that the Divinity continually geometrizes; meaning thereby, no doubt, that the laws by which the universe are governed are in accordance with the doctrines of mathematics. It does not appear that Plato composed any work expressly on the mathematics; but his invention of the geometrical analysis may be considered as one of the greatest improvements the science has received. The theory of Conic Sections originated in the Platonic school; and a third discovery was that of Geometrical Loci, a theory not only beautiful and interesting on account of the abstract truths which it unfolds, but also by reason of its great importance in the resolution of geometrical problems. The celebrated pro- blem concerning the duplication of the cube was agitated about his time; but its origin was earlier; for Hippocrates, as has been already stated, had reduced it to the finding of two continued mean proportionals. Plato himself gave a solution of the problem; and it was also resolved by Archytas, Eudoxus, Eratosthenes, and Menaechmus. The solutions of eleven of the ancient geometers have been preserved in the Commentary of Eutocius on Archimedes' Treatise concerning the Sphere and Cylinder. It is probable that the trisection of an angle, a problem of the same degree of difficulty as the duplication of the cube, excited the efforts of the Platonic school. We have indeed no positive testimony in favour of this conjecture; but the progress of the human mind does not allow us to doubt that the problem was of great antiquity. After the solution of the very simple problem, the bisection of an angle, it was natural to think of dividing it into three equal angles; and those who first thought of the latter problem would no doubt be astonished to find, that two problems so like in appearance should yet differ so much in the difficulty of their solution. The trisection of an angle, and the duplication of a cube, are in fact problems of such a nature that they cannot be resolved by straight lines and circles alone, the only lines admitted by the ancients into their elements of geometry. The reason of this impossibility would not be apparent to the ancient geometers, although it is made evident by the modern doctrines of algebra.

The names of the individuals to whom we owe the particular discoveries which have been specified have not come down to us; they must therefore be regarded as belonging to the Platonic school generally. Proclus has recorded the names of a number of the geometers of that school. Some of advanced years attended it as friends of its celebrated head, or from respect to his doctrines; others, chiefly young persons, as disciples. Of the first class were Laodamus, Archytas, and Theaetetus: Laodamus was one of the earliest to whom Plato communicated his method of analysis before he made it public, and he is said to have profited much by the possession of this instrument of discovery. Archytas was a Pythagorean, having extensive knowledge in geometry and mechanics. He frequently visited Plato at Athens; but in one of his voyages he perished by shipwreck. Theaetetus was a rich citizen of Athens, who had studied along with Plato under Socrates. He appears to have cultivated the theory of the regular solids, called now the Platonic bodies.

The progress of geometry now required that its elements should be new modelled. This was done by Leon, a scholar of Neoclis, or Neoclides, a philosopher who had studied under Plato. To Leon has been ascribed that part of the solution of a problem called its determination, which treats of the cases in which the problem is possible, and of those in which it cannot be resolved. Eudoxus of Cnidus was one of the friends and contemporaries of Plato; he generalized many theorems, and thereby greatly advanced geometry. It is believed that he cultivated the conic sections; and even the invention of these has been ascribed to him. He resolved the problem of the duplication of the cube; and it is to be regretted that Eutocius, who speaks with contempt of his solution, has not recorded it with others in his Commentary on Archimedes. Diogenes Laertius has attributed to him the invention of curve lines in general; from which we may infer, that other curves besides the conic sections were known in the school of Plato. Archimedes, in his Treatise on the Sphere and Cylinder, says that Eudoxus found the measure of the pyramid and cone, and that he was especially occupied with the contemplation of solids. Some have given him the credit of being the writer of the fifth book of Euclid, which treats of proportion. Amongst the numerous mathematicians of the Platonic school, there was one, Cratistus, who is particularly mentioned by Proclus. He was a remarkable person; for his knowledge of his science was in a manner innate. There was no problem of that time, however difficult, which he could not resolve by his natural geometry. He was the Pascal of antiquity. There were two brothers, Menaechmus and Dinostratus, who also greatly distinguished themselves. The former, a particular disciple of Plato, extended the theory of the conic sections. Eratosthenes seems to attribute to him the merit of having invented them; at any rate, he was the first who solved the problem of doubling a cube by means of these curves. Dinostratus followed in the path of his brother, and advanced geometry by his discoveries. He is particularly known as the reputed inventor of a curve called the quadratrix, by which he seems to have intended to divide an angle or an arc of a circle in any given ratio. The curve takes its name from a property by which, could it be constructed geometrically, the quadrature of the circle would be obtained. According to Pappus, Dinostratus discovered this property of the curve; and probably this is the reason why it has been called the quadratrix of Dinostratus, just as we say the spiral of Archimedes. Indeed there is some ground for believing that the curve was invented by Hippas, a philosopher and skilful geometer, contemporary with Socrates.

It was chiefly in geometry that the followers of Plato excelled. In this they imitated their leader, who directed his attention to abstract speculations, rather than to the study of nature. The conic sections, one of their most refined and profound geometrical theories, remained merely an intellectual speculation from its first invention till the days of Kepler and Newton, when it acquired a high importance on account of the discovery that the orbits of the planets are conic sections.

The school of Plato, after the death of its founder, was divided into two others, which, although opposed to each other on various points, yet agreed in holding the mathematics in esteem, and in regarding them as an indispensable preparation for the study of philosophy. It is recorded, that a person ignorant of geometry and arithmetic having presented himself for admission into the school of Xenocrates, the successor of Plato after Speusippus, he was repelled by its chief, with an intimation that he was not properly prepared for the study of philosophy: Anno philosophorum non habes, said the philosopher.

The mathematics, thus encouraged and protected, continued to be cultivated and improved. The celebrated geometer Euclid was of this school; and it may be conjectured, from the age in which he lived, that he had acquired his knowledge under the tuition of the first successors of Plato. So also, we may presume, was Aristaeus, another celebrated geometer of antiquity, though little known in our time, because his works have perished. We learn, however, from Pappus, that he was one of the ancients who contributed the most to the progress of the sublime geometry. He was the author of two excellent works; one a treatise on The Conic Sections, in five books, which contained a great part of what Apollonius has given in the first four books of his work; the other, also in five books, treated of Solid Loci. Pappus, in prescribing a course of study to his son, places this last work next in order after the Conics of Apollonius. Hence we may infer that the propositions which it contained were of a higher order of difficulty, and required a previous knowledge of the conic sections. Euclid entertained great esteem for Aristaeus. We may hence conclude that he was either his disciple or his intimate friend.

The pure mathematics were less cultivated in the school of Aristotle than they had been in that of Plato; they were not, however, neglected; for, unlike the modern peri-