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the others being not near so necesary to be known.

But these Elements of Barrow's hithera to published, notwithstanding the Brevity and Confpicuity of the Demonftrations, which renders them preferable to any others, are subject to come Deficiencies and Faults. Particularly the Schemes of the Propositions, mere. Copies of those in Peter Herigon's EUCLID, are in general too Small, and indistinit ; and many ill adapted to the generality of the Propositions. And others again, almost unintelligible, as those of Prop. 29, and 30. Lib. II. and Prop. 17. Lib. 12.

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Moreover, the second Book wants the Schemes of Prop. 5, 6, 7, 8, which by a bare Contemplation and Attention to the Text, would be almost fufficient to evince their Truth, without reading the Demon, ftrations,

There These Inconveniencies I have obviated, 'in the following Sheets, where the Schemes are made large and distinct, adapted to the generality of the Propositions; the Lines drawn for Construction dotted, to distinguish them from given Lines: others easy to be comprehended put instead of those of Prop. 29, and 30. Lib. 11. and Prop. 17. Lib. 12. and those of Prop. 5, 6, 7, 8. Lib. 2. wanting, are here supply'd,

I have likewise left out fome, and alter'd others of the Algebraick Demonstrations of the Second Book, which appeared to me too Intricate for a Learner not used to that Mee thod, and

substituted more easy ones in their, room. I have also adapted other Demon, strations to the Schemes of Prop. 29, 30. Lib. 11. and Prop. 17. Lib. 12. and have so distinguished the Schemes representing the Planes and Solids of the Eleventh and Twelfth Books, that a Learner's Imagination will be almost as much asisted as if he had real Material Planes and Solids to viera


Not long after the firft Publication of these Elements in Latin, å bad English Translation came out by an unknown hand, who was ignorant of the Subject, as, plainly appears in Def. 1. Lib. I, where he says that a Line is Longitude without, Latitude. And in Def. s. where he again repeats the words Longitude and Latitude for Length and Breadth. And in Prop. 1. Lib. 5, &c.

Tet notwithstanding, this Translation has been reprinted more than once, without any Correction or Alteration, not fo much as to make it just and tolerable

, English, which obliged me to the Trouble of new doing the following Books, and altering them as above related, not doubting of their acceptance by the English Reader,

I have one thing more to say, which is, That it is much better to have the Schemes of the Propositions in the same Pages with the Propositions (as they are in this Tract).


fingly, than to have many of them together in one Cut *, because the Learner's Attention to the Proposition he is reading, will be interrupted not only by constantly taking his Eye off from the

place he is reading, to view the Scheme, which will always be too distant, let the Cuts fold out never so advantageously; but likewise by the other circumjacent Figures of the same Cut, not to mention the Trouble of finding a Figure Sometimes. Nay, evena Mathematician of a Languid Tafte, will lay a Book aside, rather than take the trouble of seeking out the Figure of a Proposition among a num. ber all together in one Cut.

* As in Taquet's Euclid, and the English Edition of Keil's Commandine's Euclid,

An EXPLANATION of the Notes of

Characters used in this Treatise.

Signifies Equality, as A=B, or AB = BC; or AB=BC=CD; implies that A is equal to B, or AB equal to BC, or AB equal to BC equal to CD. Signifies Majority, as

AB, or AB = BC; implies, that A is greater than B, or AB greater thản BC.

Signifies Minority, as A, B, or AB BC; signifies, that A is less than B, or AB less than BC.

Signifies, that the Quantities between which itis, are added or to be added, as A+B=C+D, implies that A added to B, is equal to C added to Di and AB+CD = EF + GH JIK, fignifics that AB added to CD is equal to EF added to GH added to IK.

Signifies Subftraction, or that the latter of the two Quantities it is between, is subitracted from the former, as A - B or AB-CD, implies, that B is fubftraéted from A, or CD from AB that is, A - B or AB - CD is the Difference of A and B, or of AB and CD.

* Is the sign of Multiplication, as A X B, or AB X BC, or AB x BC CD signifies, that A is multiply'd or drawn into B, or AB multiply'd or drawn into BC, or AB multiply'd by BC, inultiply'd by CD. The conjunction of the Letters fignifies the same thing, 28 A x B=AB, or ABX BC=ABC.

✓ Signifies the side of a Square, as VAB is the Square Root, or the side of the Square AB. And


z over one or more Quantities, fignifies the Square or Cube of them, as Ā* or or Aba or AB + AC signifies the Square of A, or of AB, or of AB + AC. Underftand the fame of the Cubes.

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