SIR,If you think the above drawing of a Machine for destroying that obnoxious reptile, the Bug, and ridding houses of it in a clean, efficacious manner, it is very much at your service for the benefit of all who are troubled with this domestic affliction. I have destroyed, and cleaned my bedsteads and house of these disagreeable inmates in the course of an hour and a half, without inaking any slops or soiling the least thing. I think it would be well if innkeepers, &c. would have a machine of this sort always in their house-the expense is only 2s. 6d. I am, Sir, yours, &c. G. BROWN. thing coming in its way. It can be carried by the handle, and the spout placed any where that bugs are supposed to harbour. I assure you, Sir, though this simple contrivance may be laughed at, it is, nevertheless, as useful a piece of furniture as any in a house where these noxious vermin are troublesome. MR. DICKINSON'S CLEANSING APPARATUS. SIR,-In answer to X. X. (page 78, vol. Iv.) I beg to state that I have never used my cleansing apparatus as a substitute for the gyle tun, but cleanse from the tun much earlier than the scientific brewer lays down his law for, and have not found any ill effect from a high fermentation. I never use a cover to the cleansing cap. Should X. X. be in London, I shall be happy to show him, or any person favouring me with a call, a very considerable improvement on its conveniency. I am at a loss to know how the word clearing should have been substituted for cleansing, in all publications in which the apparatus has been noticed. Your well-wisher, R. W. DICKINSON. Tything, Worcester, May 21. 124 BRIEF ACCOUNT OF GEOMETRY. BRIEF ACCOUNT OF GEOMETRY. BY JAMES ELMES, ESQ. M.R.I.A. From his Dictionary of the Fine Arts, now publishing in Numbers. GEOMETRY. [Geometria, Lat. TewueTpía, Gr.] In all the arts, but more especially in architecture. The science of extension, quantity, or magnitude, abstractedly considered; demanding the greatest attention from the scientific artist. "There is a certain degree of geometrical knowledge," says an able writer in Dr. Brewster's Encyclopædia, "which naturally arises out of the wants of man in every state of society. It is impossible to build houses and temples, or to apportion territory, without employing some of the principles of geometry. Hence we cannot expect to find a period of society or a country in which it was altogether unknown." Ancient writers have generally supposed that it was first cultivated in Egypt; and, according to some, it derived its origin from the necessity of determining every year the just share of land that belonged to each proprietor, after the waters of the Nile, which annually overflowed the country, had returned into their ordinary channel. It may, however, be remarked, that the obliteration of the landmarks by the inundation is quite a conjecture, and not a very proba ble one. Some writers, among whom is Herodotus, fix the origin of geometry at the time when Sesostris intersected Egypt by numerous canals, and divided the country among the inhabitants. Sir Isaac Newton has adopted this opinion in his Chronology, and has supposed that this division was made by Thoth, the minister of Sesostris, who, according to him, was the same as Osiris; and this conjecture is supported by some ancient authorities. Aristotle has, however, attributed the invention to the Egyptian priests, who, living secluded from the world, had leisure for study. Thus various opinions have been entertained respecting the origin of geometry, but all have agreed in fixing it in Egypt. The celebrated philosopher, Thales of Miletus, transplanted the sciences, and particularly mathematics, from Egypt into Greece. He was born about six hundred and forty years before Christ, and being unable to gratify his ardent desire for knowledge at home, he travelled into Egypt at an advanced period of life, where he conversed with the priests, the only depositories of learning in that country. Diogenes Laertius relates, that he measured the height of the pyramids, or rather the obelisks, by means of their shadow; and Plutarch says, that the King Amasis was astonished at this instance of sagacity in the Greek philosopher; which is a proof that the Egyptians had made but little progress in the science. It is also stated by Proclus, that Thales employed the principles of geometry to determine the distance of vessels remote from shore. On his return to Greece, his celebrity for learning drew the attention of his countrymen; he soon had disciples, and hence the foundation of the Ionian school, so called from Ionia, his native country. There were some slight traces of what may be called natural geometry in Greece, before the time of Thales. Thus Euphorbus of Phrygia is said to have discovered some of the properties of a triangle; the square and the level have been ascribed to Theodorus of Samos; and the compasses to the nephew of Dædalus. But these can only be considered as a kind of instructive geometry; the origin of the true geometry among the Greeks must be fixed to the period of the return of Thales. It was he that laid the foundation of the science, and inspired his countrymen with a taste for its study; and various discoveries arè attributed to him, concerning the circle, and the comparison of triangles. In particular, he first found that all angles in a semicircle are right angles; a discovery which is said to have excited in his mind that lively emotion which is perhaps only felt by poets and geometers: he foresaw the important consequences to which this. proposition led, and he expressed his gratitude to the muses by a sacrifice. This, however, is but a small part of what geometry owes to him; and it is mach to be regretted, that the loss of the ancient history of the science should hare left us in uncertainty as to the full extent of the obligation. It is probable that the greater number of the disciples of Thales were acquainted` with geometry; but the names of Ameristus and Anaximander onlyhave reached our times. The first is said to have been a skilful geometer; the other composed a kind of elementary treatise or introduction to geometry, the earliest on record. Thales was succeeded in his school by Anaximander, who is said to have invented the sphere, the gnomon, geographical charts, and sundials; he was succeeded by Anaximenes; and this philosopher again was succeeded by his scholar Anaxagoras, who, being cast into prison on account of his opinions relating to astronomy, employed himself in attempting to square the circle. This is the earliest effort on record to resolve the most celebrated problem in geometry. Pythagoras was one of the earliest and most successful cultivators of geometry. He was born about 580 years before the Christian era; he studied under Thales, and by his advice travelled into Egypt. Here lie is said to have consulted the Co BRIEF ACCOUNT OF GEOMETRY. lumns of Sothis, on which that celebrated person had engraven the principles of geometry, and which were disposed in subterranean vases. A learned curiosity induced him to travel also into India; and it is far from being improbable that he was more indebted for his knowledge to the Brahmins, on the banks of the Ganges, than to the priests of Egypt. On his return, finding his native country a prey to tyranny, he settled in Italy, and there founded one of the most celebrated schools of antiquity. He is said to have discovered that, in any right-angled triangle, the square on the side opposite the right angle is equal to the two squares on the sides containing it, and, on this account, to have sacri ficed one hundred oxen to express his gratitude to the muses. This, however, was incompatible with his moral principles, which led him to abhor the shedding of blood on any account whatever; and besides, the moderate fortune of a philosopher would not admit of such an expensive proof of his piety. The application which the Pythagoreans made of geometry gave birth to several new theo ries, such as the incommensurability of certain lines, for example, the side of a square and its diagonal, also the doctrine of the regular solids, which, although of little use in itself, must have led to the discovery of many propositions in geometry. Diogenes Laertius has attributed to Pythagoras the merit of having discovered that, of all figures having the same boundary, the circle among plain figures, and the square among solid figures, are the most capacious: if this was so, he is the first on record that has treated of isoperimetrical problems. The Pythagorean school sent forth many mathematicians; of these, Archytas claims attention, because of his solution of the problem of finding two mean proportionals; also on account of his being one of the first that employed the geometrical analysis, which he had learned from Plato, and by means of which he made many discoveries. He is said to have applied geometry to mechanics, for which he was blamed by Plato; but probably it was rather for applying, on the contrary, mechanics to geometry, as he employed motion in geometrical resolutions and constructions. Democritus of Abdera studied geometry, and was a profound mathematician. From the titles of his works, it has been conjectured that he was one of the prin cipal promoters of the elementary doctrine respecting the properties of circles and spheres, and concerning irrational numbers and solids. He treated besides of some of the principles of optics and perspective. Hippocrates was originally a merchant, but having no taste for commerce, his affairs went into disorder; to repair 125 them, he came to Athens, and was one day led by curiosity to visit the schools of philosophy. There he heard of geometry for the first time; and it is probable there is a natural adaptation of certain minds to particular studies; he was instantly captivated with the subject, and became one of the best geometers of his time. He also was the first that composed Elements of Geometry, which, however, have been lost, and are only to be regretted, because we might have learned from them the state of the science at that period. It has been said that, notwithstanding his want of success in commerce, he retained something of the mercantile spirit: he accepted money for teaching geometry, and for this he was expelled the school of the Pythagoreans. This offence, we think, might have been forgiven, in consideration of his misfortunes, Two geometers, Bryson and Antiphon, appear to have lived about the time of Hippocrates, and a little before Aristotle. These are only known by some animadversions of this last philosopher on their attempts to square the circle. It appears that before this time geometers knew that the area of a circle was equal to a triangle, whose base was equal to the circumference, and perpendicular equal to the radius. Having briefly traced the progress of geometry during the two first ages after its introduction into Greece, we come now to the origin of the Platonic school, which may be considered as an era in the history of the science. Its celebrated founder had been the disciple of a philosopher (Socrates) who set little value on geometry; but Plato entertained a very different opinion on its utility. After the examples of Thales and Pythagoras, he travelled into Egypt, to study under the priests. He also went into Italy to consult the famous Pythagoreans, Philolaus, Timæus of Locris, and Archytas, and to Cyrene to hear the mathematician Theodorus. On his return to Greece, he made mathematics, and especially geometry, the basis of his instructions. He put an inscription over his school, forbidding any one to enter that did not understand geometry; and when questioned concerning the probable employment of the Deity, he answered, that he geometrized continually, meaning, no doubt, that he governed the universe by geometrical laws. It does not appear that Plato composed any work himself on mathematics, but he is reputed to have invented the geometrical analysis. The theory of the conic sections originated in this school; some have even supposed that Plato himself invented it, but there does not seem to be any sufficient ground for this opinion. These discoveries must be attributed 126 BRIEF ACCOUNT OF GEOMETRY, to the Platonic school in general; for it is impossible to say with whom each orinated. Some of advanced years frequented the school as friends of its celebrated head, or out of respect for his doctrines; and others, chiefly young persons, as disciples and pupils. Of the first class were Loadamus, Archytas, and Theaetetus. Laodamus was one of the first to whom Plato communicated his method of analysis, before he made it public; and he is said by Proclus to have profited greatly by this instrument of discovery. Archytas was a Pythagorean of extensive knowledge in geometry and mechanics. He had a great friendship for Plato, and frequently visited him at Athens but in one of his voyages he perished by shipwreck. Theætetus was a rich citizen of Athens, and a friend and fellow-student of Plato under Socrates, and Theodorus of Cyrene, the geometer. He appears to have cultivated and extended the theory of the regular solids. Passing over various geometers who are said to have distinguished themselves, but of whom hardly any thing more than the names are now known, we shall only mention Menæchinus and his brother Dinostratus. The former extended the theory of conic sections, insomuch that Eratosthenes seems to have given him the honour of the discovery, calling them the curves of Menæchmus. His two solutions of the problem of two mean proportionals are a proof of his geometrical skill. Several discoveries have been given to Dinostratus; but he is chiefly known by a property which he discovered of the quadratrix, a curve supposed to have been invented by Hippias of Elis. The progress of geometry among the Peripatetics was not so brilliant as it had been in the school of Plato, but the science was by no means neglected. The successor of Aristotle composed several works relating to mathematics, and particularly a complete history of these sciences down to his own time: there were four books on the history of geometry, six on that of astronomy, and one on that of arithmetic. What a treasure this would have been, had we now possessed it! The next remarkable epoch in the history of geometry, after the time of Plato, was the establishment of the school of Alexandria, by Ptolemy Lagus, about 300 years before the Christian era. This event was highly propitious to learning in general, and particularly to every branch of mathematics then known; for the whole was then cultivated with as much attention as had been bestowed on geometry alone in the Platonic school. It was here that the celebrated geometer, Euclid, flourished under the first of the Ptolemies his native place is not certainly known, but he appears to have studied at Athens, under the disciples of Plato, before he settled at Alexandria. Pappus, in the Introduction to the seventh book of his Collections, gives him an excellent character, describing him as gentle, modest, and benign towards all, and more especially such as culti vated and improved the mathematics. There is an anecdote recorded of Euclid, which seems to show he was not much of a courtier: Ptolemy Philadelphus having asked him whether there was any easier way of studying geometry than that commonly taught; his reply was, "There is no royal road to geometry." This celebrated man composed treatises on various branches of the ancient mathematics, but he is best known by his Elements, a work on geometry and arithmetic, in thirteen books, under which he has collected all the elementary truths of geometry which had been found before his time. The selection and arrangement have been made with such judgment, that, after a period of two thousand years, and notwithstanding the great additions made to mathematical science, it is still generally allowed to be the best elementary work on geometry extant. Numberless treatises have been written since the revival of learning, some with a view to improve, and others to supplant, the work of the Greek geometer; but in this country, at least, they have been generally neglected and forgotten, and Eu clid maintains his place in our schools. Of Euclid's Elements the first four books treat of the properties of plane figures; the fifth contains the theory of proportion, and the sixth its application to plane figures; the seventh, eighth, ninth, and tenth, relate to arithmetic, and the doctrine of incommensurables; the eleventh and twelfth contain the elements of the geometry of solids; and the thirteenth treats of the five regular solids, or Platonic bodies, so called because they were studied in that celebrated school: two books more, viz. the fourteenth and fifteenth, on regular solids, have been attributed to Euclid, but these rather appear to have been written by Hypsicles of Alexandria. Besides the Elements, the only other entire geometrical work of Euclid, that has come down to the present times, is his Data. This is the first in order of the books written by the ancient geometers to facilitate the method of resolu tion or analysis. In general, a thing is said to be given, which is actually exhi bited, or can be found; and the propositions in the book of Euclid's Data show what things can be found from those which by hypothesis are already known, In the order of time, Archimedes is the next of the ancient geometers that has drawu the attention of the moderns. He was born at Syracuse, about the year 287 A. C. He cultivated all the parts of BRIEF ACCOUNT OF GEOMETRY. mathematics, and in particular geometry. The most difficult part of the science is that which relates to the areas of curve lines and to curve surfaces. Archimedes applied his fine genius to the subject, and he laid the foundation of all the subsequent discoveries relating to it. His writings on geometry are numerous. We have, in the first place, two books on the sphere and cylinder; these contain the beautiful discovery, that the sphere is two-thirds of the circumscribing cylinder, whether we compare their surfaces or their solidities, observing that the two ends of the cylinder are considered as forming a part of its surface. He like wise shows that the curve surface of any segment of the cylinder, between two planes perpendicular to its axis, is equal to the curve surface of the corresponding segment of the sphere. Archimedes was so much pleased with these discoveries, that he requested, after his death, that his tomb might be inscribed with a sphere and cylinder. Eratosthenes was another great geometrician, and flourished in the Alexandrian school, about the time of Archimedes. He was born 276 A. C., and, as a geometer, ranks with Aristæus, Euclid, and Apollonius. About the time that Archimedes finished his career, another great geometriclan appeared,named Apollonius of Perga, born 240 A. C. He flourished principally under Ptolemy Philopater, or towards the end of that century. He studied in the Alexandrian school under the successors of Euclid; and so highly esteemed were his discoveries, that he acquired the name of the Great Geometer. The names of several other great mathematicians of antiquity, contemporary with Archimedes and Apollonius, have come down to us; but they are more referrible to a distinct work on geometry alone, which is of too much importance to be condensed into a single article of a work like this. We must, therefore, refer our readers, who would inform themselves properly on this important guide to all excellence in art or science, to the following works: On the history of geometry, to Montucla, "Histoire de Mathematiques," second edition. Bossut's "General History of Mathematics," of which there is a good English translation. Dr. Hutton's "Mathematical Dictionary," second edition, 4to. Lond. 1815. Dr. Brewster's Edinburgh Encyclopædia," to which we are much indebted in this article. The Encyclopædia Metropolitana," and similar works. On the elements and practice of geometry, "Euclid," of which there are many editions; the first is that of Ratdoit, 1482. Dr. Barrow's edition of all the books, and the "Data," and Dr. Horsley's of the first twelve, from the 66 127 Latin versions of Commandine and Gregory, and the "Data," are among the most valuable. "Archimedes ;' the best edition, of which are Torelli's, in Greek and Latin, Oxford, 1792; and Peyrard's French translation, Paris, 1808. The first edition of the Greek text was that of Venatorius, in 1544. Apollonius; all the writings that have been recovered of this celebrated geometer are-1. "The Section of a Ratio;" and, 2. "The Section of a Space," which were restored by Snellius, 1607; and by Dr. Halley, in 1706. 3. Determinate Section;" Snellius restored these in his "Apollonius Batavus," 1601. There is an English translation by Lawson, to which is added a new restoration, by Wales, 1772. Simson has restored this work in his "Opera Reliqua, 1776; and Giannini, an Italian geometer, in 1773. 4. "Tan gencies;" Vieta restored this in his Apollonius Gallus," 1600. Some additions were made by Ghetaldus, and others by Alexander Anderson, in 1612. The labours of Vieta and Ghetaldus have been given in English by Lawson, 1771. 5. "The Plani Loci ;" these have been restored by Schooten, 1656; and Fermat, 1679; but the best restoration is that of Dr. Simson, 1749. 6. "The Inclinations;" these were restored by Ghetaldus, in his "Apollonius Redivivus, 1607: to these there is a Supplement by Anderson, 1612; a restoration by Dr. Horsley, 1770; and another by Reuben Burrow, 1779. Theodosius and Menelaus, 1558, 1675, and an Oxford edition by Hunter in 1707. Proclus," Commentarium in primum Euclidis Librum, libri iv. Latine vertit." F. Baroccius, 1560. Proclus has also been ably translated by Taylor, 1788. Eratosthenes's "Geometria," &c. cum annot. 1672. Albert Durer, Institutiones Geometricæ, 1532. Kepler, "Nova Steriometria,' &c. 1618. Van Culen, "De Circulo et Adscriptis," 1619. Des Cartes, "Geometrie," 1637. Toricelli, "Opera Geometrica," 1644. Oughtred, "Clavis Mathematica," 1653. James Gregory, "Geometria Pars Universalis," 1668. Barrow, "Lectiones Opticæ et Geometrica," 1674;" Lectiones Mathematicæ," 1683. David Gregory, "Practical Geometry," 1745. Sharp, "Geometry Improved," &c. 1718. Stewart, "Propositiones Geometricæ," 1763. Thomas Simson," Elements of Geometry," 1747 and 1670. "Select Exercises," by the same, 1752. Emerson's "Elements of Geometry," 1763. Lacroix, "Elémens de Géometrie Descriptive," 1795. Playfair, "Origin and Investigations of Porisms," Edin. Trans. vol. iii. Legendre's "Élémens de Géometrie," ninth edition, 1812. Leslie, "Elements of Geometry, Geometrical Analysis, and Plane Trigonometry," second edition, 1811, |