GEOMETRY. 



401 



And this arte may be made on the earth or on white 

 paper, or uppou any other thing whereon it may coni- 

 nuxliously be done, so that the prickes and lines may 

 be knowen. The beginning and original of this arte 

 came from the Indians, which found it before the 

 world was drowned. It may be practised whenso- 

 ever that a man will, according to the demand that 

 is made, be it night or day, fair weather or fowle, 

 raine or winde." One of the oldest writers on geo- 

 mancy is said to be Philo Gudaeus. Cornelius Agrippa, 

 besides some notices in his work De occulta Philoso- 

 phia, has left an express tract, De Geomantia, of 

 which he speaks with much honesty in a production 

 of his later years, De Sanitate Scientiarum: " I have 

 written also a certain book of geomancie, far differ- 

 ing from the other, but no lesse superstitious, false, 

 or, if youlyst, I wyllsay, lying." (Sandford's trans- 

 lation, 1575.) In a subsequent chapter (36), he dis- 

 tinguishes two sorts ofgeomancy: "All they which 

 write hereof do affirme, that geomancie is the daugh- 

 ter of astrologie, whereof we have spoken in arith- 

 meticke, which fashioneth certain figures attributed 

 to the heavenly signes by which they divine. There 

 is also another kind of geomancie which Almadul the 

 Arabian introduced and brought in, the which doth 

 divine by certain conjectures taken of similitudes of 

 the cracking of the earth, of the moving, cleaving, 

 swelling, either of itselfe, or els of inflammation and 

 heate, or of thundrings that happen, the whiclie also 

 is grounded upon vaine superstition of astrologie, as 

 that which observeth houses, the newe moones, the 

 rising and forme of the starres." This science was 

 flourishing in the days of Chaucer, and was deeply 

 cultivated by Dryden, at the time of his rifaccimento 

 of the Knight's Tale. Cattan, whose book we have 

 already mentioned, appears to have been very largely 

 employed. Among other figures, he presents us with 

 one cast for the lord of Ferte, when he was in love 

 with my lady Bye ; one for the lord of Lymoges, to 

 know whether a musician, who had absconded from 

 his service, would return ; one for my lord Clerinout 

 of Lodeves, respecting his litigated inheritance ; some 

 relative to the sale and purchase of horses ; one to 

 determine whether the cardinal Trivulfee (Trivulzio] 

 should succeed in making peace between the king o: 

 France and the emperor ; one to determine the day 

 on which the emperor should quit Nice ; another to 

 ascertain whether the count of Novelaire was dead or 

 alive ; a figure to find the question for which another 

 figure, found by accident, was made ; others to dis- 

 cover people's thoughts, or to find out their names. 

 It may be gratifying to our readers to know, tliat this 

 science is "no arte of inchaunting, as some may 

 suppose it to be, or of divination which is made by 

 diabolike invocation ; but it is a part of natural 

 magicke, called of many worthy men the daughter of 

 astrologie, and the abbreviation thereof." There is 

 a tract on geomancy by Bartolomeo Cocle, who 

 styles himself Filosofo integerrimo (Venice, 1550). 

 Oughtred, who died in 1660, appears to have been 

 one of the latest serious cultivators of geomancy. 



GEOMETRY (from the Greek, signifying the art 

 of measuring land) ; the branch of pure mathematics 

 which treats of the magnitudes of dimensions. It is 

 divided into longimetry, occupied exclusively with 

 lines, planimetry, occupied with planes or surfaces, 

 and stereometry, treating of solid bodies, their con- 

 tents, &c., and the doctrine of the functions of the 

 circle, and its application to certain figures, formed 

 by lines, from which originate (a.) trigonometry, (b.) 

 tctragonometry , (c.) polygonometry , (d.) cyclometry, 

 which teach us to find, from the dimensions of certain 

 parts of a figure, those of certain other parts, by 

 which particularly altitudes and depths are to be 

 measured. Geometry is divided into elementary and 

 m. 



applied. The former, or theoretical geometry, treats 

 of die different properties and relations of the magni- 

 iudes of dimension in theorems and demonstrations, 

 which the latter applies to the various purposes of 

 ife in problems and solutions. Geometry is taught 

 .n different ways ; as, for instance, by diagrams, 

 which is called constructive geometry, or by the appli- 

 cation of algebra to dimension, which is called analyti- 

 cal geometry. The invention of this important science 

 ascribed by some to the Chaldeans and Babylon- 

 ians ; by others to the Egyptians, who were obliged 

 to determine the boundaries of their fields, after the 

 inundation of the Nile, by geometrical measurements. 

 According to Cassiodorus, the Egyptians either 

 derived the art from the Babylonians, or invented it 

 after it was known to them. Thales, a Phoenician, 

 who died 548 B. C., and Pythagoras of Sanios, who 

 flourished about 520 B. C., introduced it from Egypt 

 into Greece. The discovery of five regular geome- 

 trical bodies, thecube, tetraedron, octaedron, icosaedron 

 and dodecaedron, is ascribed to the latter. He dis- 

 tinguished himself particularly by the invention of 

 the theorem, which is called from him the Pythagor- 

 ean, and, on account of his important improvements, 

 lias received the name of magister matheseos. In 

 elementary geometry, Euclid of Alexandria is parti- 

 cularly distinguished. About a hundred years after 

 him, Archimedes extended the limits of geometry by 

 his measure of the sphere and the circle. Aristasus, 

 and, at a later period, Apollonius of Perga (who 

 flourished 260230 B. C.), did much for the higher 

 geometry. In Italy, where the sciences first revived, 

 after the dark ages, several mathematicians were 

 distinguished in the sixteenth century ; the French, 

 and, particularly, the Germans, followed. Justus 

 Byrge laid the foundation of logarithms, and, accord- 

 ing to some, was the inventor of the proportional circle; 

 others ascribe the invention to Galileo. Reinerus 

 Gemma Frisius, who died in 1555, invented the in- 

 strument used in surveying, called the plain table. 

 Simon Stevin of Bruges applied the decimal measure 

 to geometry. In 1635, Bonavent. Carallieri opened 

 the path to the higher geometry of infinites ; and, in 

 1684, Leibnitz advanced the science by the invention 

 of the differential calculus, and Newton by the theory 

 of the fluxions. Robert Hook, who died in 1703, 

 was the first who considered the influence of the re- 

 fraction of light in measuring heights. Ludolph of 

 Ceuln, or Cologne, who died at Leyden in 1610, dis- 

 covered the proportion between the diameter and the 

 circumference of the circle. In recent times, the 

 French have been most distinguished in geometry, 

 and have produced the best elementary works for 

 schools in this branch ; as, for instance, those of 

 Legendre and Monge. The Germans have a num- 

 ber of elementary works on geometry, some of which 

 are excellent. Among the most approved modern 

 works on the elements of geometry, are those of 

 Euclid, as translated by Simson, Ingram, and Play- 

 fair, and the treatises of professor Leslie, and M. 

 Legendre, above mentioned. 



1 2 3 





Definitions. A point is that which has position, 

 but no magnitude, nor dimensions ; neither length, 

 breadth, nor thickness. A line, fig. I, is length, 

 without breadth or thickness. A surface or super- 

 ficies, fig. 2, is an extension or a figure of two 

 dimensions, length and breadth ; but without thick- 

 2 41 



