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1. How the Area of a Square is obtained.
3 2. Of any Rectangle whatever. 3. Of the Triangle.
4 4. All Figures, having equal circuit, are not equal in Area. 4 5. A Square contains a grea er Area than any other
5 Rectangle, the sums of whose Sides are equal. 6. The Difference between a Rectangle and other Parallelogram 6 7. How the Areas of regular Polizons are obtained.
7 8. Of the Circle.
7 9. The Affinity between Circles, and Poligons.
8 10. That a Circle has a greater Area than a Square, or
any other Figure, having equal circuit. 11. The Area of a Rectangle ascertained, and accounted for. 12. By Duodecimals, or Feet and Inches. 13. By decimal Parts; the difference explained.
14 14 and 15. The Construction and use of Scales, for mea. 15
suring or delineating; in Decimals and Duodecimals. 16 16. Of irregular Figures; as a Field, &c.
17 17. How to divide any right-lined Figure into two equal
18 parts, by a Right Line, from any Point in any side. 18. How to obtain the Area of a Triangle, trom the 1 measure of its Sides only, without a Perpendicular.
1. How the Area of a Cube is obtained.
28 29 30
240.- 7. B. for equality of Rarios, Page. Line.
read, Ratio of Equality. 23.-9. Art. 4. for, Circles, reaa, 244.-- 2 and 4. B. for, Ratio, read, Circumferences.
Increase. 38.- 8. for, CDAB, read, CDbB. 244. 10. B. for, A &C... A&B. 41.-10. read, intersecting at F. 5. B. read, if A be 3. 42. 7. B. far, A, read, H. 257.711. for, Def. 6. read Ax. 6. 47.- 7. for., CDG, read, CDG.
-19. for, A:B, read, if A:B. 31.- Line last, for 10, Tead, 20. 7. B. far, B. reait, c, and 57. 8. B. for, 20..1. read, 18.1.
for C, read, B. $8.-16. fór, AHCD, r. GHCD. 258. 5. B. f. Axiom, r. Postulate. 62. 9. B. for, 12. 2. read, 12.3.261.-10. B. for, as A to B, read, 67.- 4. read, to the fourth.
as B to A. 70 -15. for, AH, read, Af. 267-412. B.for, D to G, r. Cto G. 75.-12. for, AC, read, a C.
268.- 2. Den. for, A and B, 96.-11. for, MN, read, KM.
read, A and C. 98.-11. for, Def. 15. and 18.7th. 300.- 4. read, Triang!es.
read, 1 and 6. 8th. 329.-10. for, 8. 4. read, 9. 4. 104.- 7. Prob. 7 for, H, read, K; 338. 6. for, AC, read, BC.
and, for, K, read, H. 339. 4. B. for, ABC, read, AB, 107.-16. for, and, read, i.e. chatis. 10. B. for, EG, read, EF.
347.-10. Dem. read, Draw otheç
355.- 6. B. for, F&G, r. D& E.
8 & 10. B. for, G, read, D. 116.-3 & 4. for, E, read, A. -Bottom, for, ECB,'r. DCB. 356.- 8. for, AG, read, CG.
358.- 3. Dam. for, BK, read,BL, 125.- 5. for, Ax. 3. read, 7.
365.- 6. for, Ax. 4. 5. read, 5.5. 135. 5. Theo. 4. .f.BAG,r.EAG.
5. for, Def. 7. read. 8. 146. 7. for, CD, read, BC. *47.-13. for, Hyp. AB, read, AC. 39.- 3. for, Ax. 11.7.read, Def. 156. 9. Th. 2. for, A E, read, AD 40.- last. for BF. read, Bf. -14. for, Ax. 3. read, 2.
408.-13. Dem, add, and having 158. 2. B. for EA, read, EH.
equal Altitudes. 171,- 6 for, EB, read, EC. 172 - 5. Bottom, for, 13th,read, 14
In the Appendix. ... 182.–13. for, Def. 44, riad, 43. 183
7. Cor.2. for, ind, r. in H. 190.- 7. Bottom, read, DG in D. 13.- 5. for, BE. read, DE. 395.-11. read, sroís each other.
14.- 8. B. for, 425, read 37€. 223.- 5. B. for AFO=FB read, 19.-19&20, for, K, read, F.
AFO= AE 0+EFO 25 -13, 15, &16, for, G, read, E. 225.– 8. Cor. read, Duodecagon. / 32. 6. for, Chord of go, read, 60,
THE mathematical World will, I doubt not, be surprizei 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 suspend their opinion, and not pass a too hafty censure, on account of the obscurity of the Author, till after they have given it a fair and candid perusal, and then proceed to judgment with candour and impartiality.
I do not pretend to much knowledge in the Mathematics, having been brought up in a way of life, very different from my inclination; yet, what time I could spare from bufiness and the demands of my family, I chose to employ in such studies; and have, by dint of study, only, and without any other instruction, made some progress in mathematical Sciences ; of which, Geometry is the first, and a sure key to the reft.
Since I have made myself, by self-application, a Proficient in Geometry, and have made some branches of the Mathematics my Study and Profession, I have often been surprized at the negligence and deficiency of our common Schools, for the cultivation of Youth who are intended to fill the middle sphere of Life, in mechanic Trades, &c. They, almost in general, pursue one common Plan or track of Learning. After the first and necessary branches, Reading, Writing, and Arithmetic; which, indeed, might be acquired in half the time it usually is; the next ftep (if the Pupil has made a progress thro’ Arithmetic in any reafonable sime) is the Grammar of the Latin Tongue through which, he sweats and labours to little purpose. If the Pupil has three or four years to spare, before he goes out to business, he perhaps gets into the Cordery or Erasmus ; 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 purpose. For, what has methanic 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 use in his Profession; even suppose he had made a tolerable proficiency, it could answer no purpose but to set 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 Mensuration, &c. were introduced in all public, common Schools, I would ask any person, who has considered these things, and their uses in Lise, which is most likely to turn to the Pupil's advantage? Is there a mechanic Profession in which Geometry or Mensuration may not be of some use ? in some particular ones it is well known to be of the greatest, 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 such Trades as relate particu. larly to Building, in general. Had fome Builders, whom I have known, been conversant in the Mathematics, or only in plane Geometry; instead 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 such, I have been at the trouble of composing this Treatise. If only the practical part is well inculcated, it will be of more service, in common Life, than a proficiency in Latin can possibly 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 consider, what particular Profession he either wishes or is destined for. In choosing of which, regard ought particularly to be had to the Boys genius and disposition, which will, ere this, be discernable. But, at all adventures, instead of flogging and driving a useless dead Language into a stupid Boy, which only renders him more so, let the practice of Geometry be introduced in its stead, in every common School; there is something entertaining
to the Mind, more than in burdening the memory with As in prefenti, and other rules of the Latin Grammar.
Accustoming Boys, early, to handle the Compasses and other drawing utensils, 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 scon have a relish for, and then they would proceed with pleasure : besides, it is an introduction to Drawing. A foundation being once laid in Geometry, they are then qualified to pursue any other branch of the Mathematics, suitable to the Profession they are intended for ; such as Mensuration, Trigonometry, Navigation, Gunnery, Fortification, Architecture, naval or domestic, Surveying, &c. In short, all the useful and necessary Employments, in the mechanic Arts, have their foundation in this most necessary Science ; which, being acquired, will, most probably, make its Poffeffor strike out of the common and vulgar track, and make him eminent and distinguishable, in whatever Profession he is casually fixed in ; as he will have laid a solid and permanent foundation, in Theory ; whereon, may, very probably, be erected, a lasting monument to his future Fame.
I have perused several Authors on the Subject, and find, that some have treated it in a manner scarce intelligible to a beginner, unless he has some knowledge of Algebra ; others would be better understood and approved, if they did not dwell too much on self - evident Propositions, which are, in themselves, perfect Axioms. Perhaps, I fhall be blamed for censuring, as useless, several Propofitions in that famous Geometrician, Euclid; but
I cannot conceive of what use is all that tedious round about method, in the 2nd Problem, Book 1. viz. “ To
put a right Line, a: a given point, equal to a given Line,” unless some particular direction, in respect of the other Line, was also given.
In Problem 3d. where the 2nd is applied, I ask, for what use? and why, after having taken the line Cin the Compasses, as Radius,