the others being not near fo necessary to be known.

But thefe Elements of Barrow's hither to published, notwithstanding the Brevity and Confpicuity of the Demonftrations, which renders them preferable to any others, are fubject to some Deficiencies and Faults. Particularly the Schemes of the Propofi tions, mere Copies of those in Peter Herigon's EUCLID, are in general too Small, and indiftinct; and many ill adapted to the generality of the Propofitions And others again, almost unintelligible, as thofe of Prop. 29, and 30. Lib. II. and Prop. 17. Lib. 12.

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

Thefe

Thefe Inconveniencies I have obviated, in the following Sheets, where the Schemes are made large and diftinct, adapted to the generality of the Propofitions; the Lines drawn for Construction dotted, to diftinguish 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 thofe of Prop. 5, 6, 7, 8. Lib. 2. wanting, are here fupply'd.

I have likewife left out fome, and alter'd others of the Algebraick Demonftrations of the Second Book, which appeared to me too Intricate for a Learner not used to that Method, and fubftituted more eafy ones in their room. I have also adapted other Demonftrations to the Schemes of Prop. 29, 30. Lib. 11. and Prop. 17. Lib. 12. and have fo diftinguished the Schemes representing the Planes and Solids of the Eleventh and Twelfth Books, that a Learner's Imagination will be almost as much affifted as if he had real Material Planes and Solids to view

Not

Not long after the firft Publication of thefe Elements in Latin, a bad English Tranflation came out by an unknown hand, who was ignorant of the Subject, as, plainly appears in Def. 1. Lib. 1. where he fays 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 fay, which is, That it is much better to have the Schemes of the Propofitions in the fame Pages with the Propofitions (as they are in this Tract)

fingly,

fingly, than to have many of them together in one Cut*, because the Learner's Attention to the Propofition 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 diftant, let the Cuts fold out never fo advantageously; but likewife by the other circumjacent Figures of the fame Cut, not to mention the Trouble of finding a Figure Sometimes. Nay, even a Mathematician of a Languid Tafte, will lay a Book afide, rather than take the trouble of feeking out the Figure of a Propofition 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

Characters used in this Treatife.

BC;

Signifies Equality, as A= B, or AB or AB BCCD; 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 thần BC.

Signifies Minority, as AB, or AB BC; fignifies, that A is lefs than B, or AB lefs than BC.

Signifies, that the Quantities between which it is, are added or to be added, as A+B=C+D, implies that A added to B, is equal to C added to D; and ABCDEFGHIK, fignifies that AB added to CD is equal to EF added to GH added to IK.

Signifies Subtraction, or that the latter of the two Quantities it is between, is fubftracted from the former, as AB or ABCD, implies, that B is fubftracted from A, or CD from AB; that is, AB or ABCD is the Difference of A and B, or of AB and CD.

T

x Is the Sign of Multiplication, as Ax B, or AB x BC, or ABX BC x CD fignifies, that A is multiply'd or drawn into B, or AB multiply'd or drawn into BC, or AB multiply'd by BC, multiply'd by CD. The conjunction of the Letters fignifies the fame thing, as A x B AB, or ABX BC ABC.

✓ Signifies the Side of a Square, as AB is the Square Root, or the Side of the Square AB.

And - 2 or 3 over one or more Quantities, fignifies the Square or Cube of them, as A or or AB or AB AC fignifies the Square of A, or of AB, or of AB+ AC. Understand the fame of the Cubes.