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1. How the Area of a Square is obtained.

2. Of any Rectangle whatever. 3. Of the Triangle.
4. All Figures, having equal circuit, are not equal in Area.
5. A Square contains a greater Area than any other
Rectangle, the fums of whofe Sides are equal.

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6. The Difference between a Rectangle and other Parallelogram 6 7. How the Areas of regular Poligons are obtained.

8. Of the Circle.

9. The Affinity between Circles, and Poligons.

10. That a Circle has a greater Area than a Square, or any other Figure, having equal circuit.

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11. The Area of a Rectangle afcertained, and accounted for.
12. By Duodecimals, or Feet and Inches.
13. By decimal Parts; the difference explained.
14 and 15. The Conftruction and ufe of Scales, for mea-
furing or delineating; in Decimals and Duodecimals.
16. Of irregular Figures; as a Field, &c.
17. How to divide any right-lined Figure into two equal
parts, by a Right Line, from any Point in any Side.}
18. How to obtain the Area of a Triangle, from the
measure of its Sides only, without a Perpendicular.

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Of Solids.

4. Of Pyramids and Cones.

1. How the Area of a Cube is obtained.

2. Of other right-angled Parallelopipeds.

3. Of any Parallelopiped, Prism or Cylinder whatever.

5. How the Area of a Sphere is ascertained.

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8. Of irregular Solids.

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A mechanichal demonstration of Theo. 20. 1.—
Line of Chords, conftructed and explained.

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Practical Geometry,

Fage. Line. 23.-9. Art. 4. for, Circles, reaa, Circumferences. 38.-8. for, CDAB, read, CDbB. 41.-10. read, interfecting at F. 42. 7. B. far, A, read, H. 47.- 7. for, CDG, read, CDG. 31.- Line laft, for 10, read, 20. 57. 8. B for, 20... read, 18.1. 58.-6. for, AHCD, r. GHCD. 62. 9. B. for, 12. 2. read, 12. 3. 67. 4. read, to the fourth. 70 -15. for, AH, read, AF. 75.-12. for, AC, read, a C. 96.-11. for, MN, read, KM. 98.-11. for, Def. 15. and 18. 7th. read, 1 and 6. 8th.

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404.-7. Prob. 7 for, H, read, K; and, for, K, read, H.

107.-16. for, and, read, i.e. that is.

In the Elements,

116.-3 & 4. for, E, read, A. $20.-Bottom, for, ECB, r. DCB. 125. 5. for, Ax. 3. read, 7. 135.5. Theo. 2.f. BAG,r.EAG. 146. 7. for, CD, read, BC. 147-13. for, Hyp. AB, read, AC. 156. 9. Th. 2. for, AE,read, AD -14. for, Ax. 3. read, 2. 258. 2. B. for EA, read, EH. 171.- 6 for, EB, read, EC.

372

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5. Bottom, for, 13th,read, 14 182.-13. for, Def. 44, read, 43. 183 7. Cor.2. for, in A, r. in H. 190. 7. Bottom, read, DG in D. 195.-11. read, rots each other. 223.- 5.

Page. Line.

240. 244.- 2

7. B. for equality of Ratios, read, Ratio of Equality. and 4. B. for, Ratio, read, Increafe. 244.-10. B. for, A&C,r. A & B. 5. B. read, if A be 3. 257.11. for, Def. 6. read Ax. 6. -19. for, A: B, read, if A:B. 7. B. far, B. read, C, and for C, read, B.

258. 5. B. f. Axiom, r. Poftulate. 261.10. B. for, as A to B, read, as B to A.

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267.-12. B. far, D to G, r. C to G. 268.- 2. DEM. for, A and B, read, A and C. 4. read, Triangles. 329.-10. for, 8. 4. read, 9. 4. 338.- 6. for, AC, read, BC. 339. 4. B. for, ABC, read, AB. 10. B. for, EG, read, EF. 347.-10. DEM. read, Draw other Right Lines.

355

356. 358. 365. 388.

349.3. DEM. for, on, read, or. 6. B. for, F&G, r. D & E. 8 & 10. B. for, G, read, D. 8. for, AG, read, CG. 3. DEM. for, BK, read, BL. 6. for, Ax. 4. 5. read, 5.5. 5. for, Def. 7. read. 8. 3. for, Ax. 11. 7.read, Def. laft. for BF. read, Bf. 408.-13. DEM. add, and having equal Altitudes.

391.

401.

In the Appendix...1

135. for, BE. read, DE. 14.- 8. B. for, 425, read 375.

-19 & 20, for, K, read, F. 25-13, 15, & 16, for, G, read, E. 32. 6. for, Chord of go, read, 60,

19. B. for AFFB,read, AFDAE O+ ENG 225.-8. Cor. read, Duodecagon.

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THE

HE mathematical World will, I doubt not, be furprized at a fresh publication of the Elements of Geometry, by one entirely unknown; and, on a plan very different from that of others who have wrote on the Subject. I hope they will fufpend their opinion, and not pass a too hafty cenfure, on account of the obfcurity of the Author, till after they have given it a fair and candid perufal, and then proceed to judgment with candour and impartiality.

I do not pretend to much knowledge in the Mathematics, ha ving been brought up in a way of life, very different from my inclination; yet, what time I could fpare from business and the demands of my family, I chofe to employ in fuch ftudies; and have, by dint of study, only, and without any other instruction, made fome progrefs in mathematical Sciences; of which, Geometry is the firft, and a fure key to the reft.

Since I have made myself, by felf-application, a Proficient in Geometry, and have made some branches of the Mathematics my Study and Profeffion, I have often been furprized at the negligence and deficiency of our common Schools, for the cultivation of Youth who are intended to fill the middle fphere of Life, in mechanic Trades, &c. They, almoft in general, pursue one common Plan or track of Learning. After the firft and neceffary branches, Reading, Writing, and Arithmetic; which, indeed, might be acquired in half the time it ufually is; the next step (if the Pupil has made a progrefs thro' Arithmetic in any reafonable sime) is the Grammar of the Latin Tongue through which, he fweats and labours to little purpofe. If the Pupil has three or four years to fpare, before he goes out to bufinefs, he perhaps gets into the Cordery or Erafmus; or, if he reaches Cornelius Nepos, he is looked on as a prodigy.

Now, it may reasonably be asked, for what purpose all this Time has been spent? which might have been employed to much better

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better purpose. For, what has mechanic Trades to do with Latin? any more than a common Porter or Carman with Logic; it may indeed complete him a Pedant or Coxcomb, but can never be of real ufe in his Profeffion; even fuppofe he had made a tolerable proficiency, it could anfwer no purpose but to fet him above his Employment, without being of any service in it.

On the contrary (supposing no particular avocation is intended for the Student) if, instead of Latin, Geometry and Menfuration, &c. were introduced in all public, common Schools, I would ask any perfon, who has confidered these things, and their uses in Life, which is moft likely to turn to the Pupil's advantage? Is there a mechanic Profeffion in which Geometry or Menfuration may not be of fome use? in fome particular ones it is well known to be of the greateft, the foundation of it; and yet, altho' the Youth was particularly intended for that Profeffion, it was, perhaps, never once so much as thought on; until, by too late experience, he finds the want of it: I mean all fuch Trades as relate particularly to Building, in general. Had fome Builders, whom I have known, been converfant in the Mathematics, or only in plane Geometry; inftead of plodding on in a low sphere of Employment, they would, if their natural, mechanical genius had been properly cultivated, have filled a more elevated station.

For the use of fuch, I have been at the trouble of compofing this Treatife. If only the practical part is well inculcated, it will be of more fervice, in common Life, than a proficiency in Latin can poffibly be. If the young Pupil has a genius, and discovers a relish for mathematical Science, let him go on with the Elements; and if he acquires a competent share of knowledge therein, it will then be time to confider, what particular Profeffion he either wishes or is deftined for. In choofing of which, regard ought particularly to be had to the Boys genius and difpofition, which will, ere this, be discernable. But, at all adventures, inftead of flogging and driving a ufelefs dead Language into a ftupid Boy, which only renders him more fo, let the practice of Geometry be introduced in its ftead, in every common School; there is fomething entertaining

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to the Mind, more than in burdening the memory with As in prafenti, and other rules of the Latin Grammar.

Accuftoming Boys, early, to handle the Compaffes and other drawing utenfils, in delineating all the Diagrams, as they proceed, will be an entertainment to them, and of great utility, rather than a perplexing study, and gradually enure them to Demonstration; which, under a proper Tutor, they would feon have a relish for, and then they would proceed with pleafure: befides, it is an introduction to Drawing. A foundation being once laid in Geometry, they are then qualified to purfue any other branch of the Mathematics, fuitable to the Profeffion they are intended for ; fuch as Menfuration, Trigonometry, Navigation, Gunnery, Fortification, Architecture, naval or domeftic, Surveying, &c. In short, all the useful and neceffary Employments, in the mechanic Arts, have their foundation in this moft neceflary Science; which, being acquired, will, moft probably, make its Poffeffor ftrike out of the common and vulgar track, and make him eminent and diftinguishable, in whatever Profeffion he is cafually fixed in; as he will have laid a folid and permanent foundation, in Theory; whereon, may, very probably, be erected, a lafting monument to his future Fame.

I have perused several Authors on the Subject, and find, that fome have treated it in a manner fcarce intelligible to a beginner, unless he has fome knowledge of Algebra; others would be better understood and approved, if they did not dwell too much on felf - evident Propofitions, which are, in themselves, perfect Axioms. Perhaps, I fhall be blamed for cenfuring, as useless, feveral Propofitions in that famous Geometrician, Euclid; but muft own, I cannot conceive of what ufe is all that tedious round about method, in the 2nd Problem, Book 1. viz. “To put a right Line, at a given point, equal to a given Line," unless fome particular direction, in refpect of the other Line, was alfo given.

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In Problem 3d. where the 2nd is applied, I afk, for what ufe? and why, after having taken the line C in the Compaffes, as Radius,

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