The Elements of Euclid: Viz. The First Six Books, Together With the Eleventh and Twelfth; The Errors, by Which Theon, or Others, Have Long Ago Vitiated These Books, Are Corrected, and Some of Euclid's Demonstrations Are Restored (Classic Reprint)

Excerpt from The Elements of Euclid: Viz. The First Six Books, Together With the Eleventh and Twelfth; The Errors, by Which Theon, or Others, Have Long Ago Vitiated These Books, Are Corrected, and Some of Euclid's Demonstrations Are Restored

5th Book, by which the Boéttine of Compound Ratios is renc dered plain and eafy. Befides, among the Definitions of the r1th Book, there is this, which is the 10th, viz. Equal and fimilar folid Figures are thofc which are contained by limilar Planes of the fame Number and Magnitude. Now, this Propofitibn is a Theorem, not a Definition; becaufe the equality of Figures of any kind mult be demonltrated, and not afl'umed; and, therefore, though this were a true Propo fition, it ought to have been dcmonfirated. But indeed this Propofition, which makes the roth Definition of the 11th Book, is not true univerfally, except in the cafe in which each of the folid Angles of the Figures is contained by no more than three plane Angles; for, in other Cafes, two folid Figures may be contained by fimilar Planes of the fame Number and Ma Jiitudc, and et be unequal to one another; as fhall be ma c evident in t e Notes fuhjoined to thefe Elements. In like manner, in the Demonfiration of the a6th Prop. Of the nth Book, it is taken for granted, that thofc folid Angles are e qual to one another which are contained by plain An les of the fame Number and Magnitude, placed in the fame rder but neither is this uniyerfally true, except in the cafe in which the folid Angles are contained by no more than three plain Angles; nor of this Cafe is there an Demonltration in the Elements we now have, though it e quite neccll'ary there fhould be one. Now, upon the xoth Definition of this Book depend the a5th and 28th propofitions of it; and, u on the nsrb and 26th depend other eight, viz. The 27th, 31 gad, 33d, 34th, 36th, 37th, 4oth of the fame Book; and the lath of the 12th Book dep ds u n the eighth of the fame, and this 8th, and the Corollary of Pi'opofition ryth, and Prop. 18th of the nth Book, depend upon the 9th Definition of the 11th Book, which is not a right Definition besaufe there may besolids contained by the fame number of fimilar plane Figures, ich are not fimilar to one another, in the true Scnfe of Simi larity received by all Geometers; and all thefe Propofitions have, for thefe Reafons, been infufliciently dcmonfirated fiuce Theon'a time hitherto. Befides, there are fcvcral other things, which have nothing of Euclid's Accuracy, and which plainly ibew, that his Elements have been much corrupted by unlkil~ ful Geometers; and, though thefe are not (0 grofs as the o thers now mentioned, they ought by no means to remain un oorre&ed. Upon thele Accounts it appeared necefl'ar and I hope will.

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