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the Angle GAB will always remain the same, and, at laft, coincide with ACD.
And if AB be moved direarly forward, to EF, still keeping aarallel to its first position, it will also coincide with EF, and the Angle GAB, or ACD, with CEF.
Hence, it is plain that the Angles x, y and z, made with GE, are all equal to one another; and, being equal to one another, the Right Lines AB, CD and EF are consequently parallel between themselves,
4. Again; since the Angles x, y and z are equal to one another; and the opposite Angles, v, u, w, are also equal between themselves, and to the other (as it was made evident in the foregoing) consequently, the Alternate Angles, v and y, u and z, &c. are equal; for, they are all equal to one another, as above (the Lines AB, CD and EF being parallel) and, conversely, if the Alternate Angles are found to be equal, the Lines, forming them, are consequently parallel.
5. Also; since a Right Line cuting another Right Line makes the Angles, on both Sides, equal to two Right Angles (by the first) it is evident, that, if the Angles , у and z are equal to one another, and the adjoining Angles x+A and y+C, &c. are each equal to two Right Angles; consequently, A, C, and E are also equal between themselves, and the internal Angles (between each two Lines) A added to y, and C added to z, are also, each, equal to two Right Angles; by the 6th. Axiom.
6. And, because A+v and y to are each equal to two Right Angles; and, À +y, on one side GE=two Right Angles; conf. v to, on the other Side GE, are, also, equal to two Right Angles. Therefore, if two parallel Right Lines, AB & CD, &c. are cut by any other Right Line, GF, the internal Angles, on each Side, are equal to two Riç hệ
Angles; and, consequently, if a Right Line cuts two Right Lines, which are not parallel (as CD and HI) they will meet on that Side where the two internal Angles (DCE+ CEI) on the fame Side, are less than two Right Angles; which, is Euclid's 121h. Axiom.
Now since all these properties of Right Lines, and of Angles formed by Right Lines, are manifest, from what has been advanced already, there is, I think, but little occasion for Demonstration ; nevertheless, in conformity to the Antients, I have formed them into Propositions, for the satisfaction of the scrupulous, and for the sake of reference, in other Works, as well as in this; yet, I must own, I am clearly of opinion, that this brief extract is sufficient, towards attaining all the necessary knowledge inculcated by the first fix Propofitions. It will, at least, be of use to the young Student, in giving him a clear Idea of the Properties therein demonstrated; which, by being more familiarized to them, will appear more evident and conspicuous; the necessary knowledge, contained in them, will be deeper rooted and more securely established, and the Pupil better prepared to proceed with the Demonstrations of the proper; tics of Right lined Figures.
use of in this and in other mathematical Trearices.
Note of Equality.
is thus read, A is equal to B; frequently, A equal B. + Note of Addition, Plus, or more.
Thus A+B, signifies the Sum of A added to B,
Note of Subtraction; Minus, or less. ·
A-B=2, must be read, A, less B, is equal to 2. х Note of Multiplication.
Thus AXB, signifies, A multiplied into B.
Note of Division.
Let A be 6 and B 2; then, A, B = 3. :::: Note of Equality of Proportion.
Thus, A:B::C:D, signifies, that A bears the same
Note of continued Proportion, or geometrical Pro-
Square. Thus, AB O, fignifies, the Square of the Line ĄB.
A-.B. А В
Rectangle. Thus, ACO; or ABC, signifies D
the Rectangle under the Lines AB and BC; or, AB X BC. And, ABCO=BDO, must be read thus; the Rect. ABC is equal to BD Square, of to the Square of BD.
Rectangles and other Parallelograms, are frequently de. noted by two Letters, at opposite Angles. As AC or DB.
2 ABC; or 2 ABO; must be read, two Rectangles ABC, or twice the Rect. AB; and, 2 or 3 AB a Square must be understood, twice or thrice the Square of AB. A Triangle. Thus, AABC, fignifies the Triangle ABC.
A X I 0 MS.
An Axiom (as already defined) is a manifest Truth, clear in itself, and therefore, does not require to be demonstrated : Nevertheless, an Illustration will not, I prefume, be thought superfluous or unnecessary.
1. Things which correspond, coincide, or mutually agree
with each other, in every Part, are equal.
Thus, A is equal B; if every part of the Figure
A coincides with B.
2. The whole is greater than a Part of the same Thing,
For it is equal to all its Parts.
3. If two, or more, Things, or Quantities, are each
equal to the same third Thing or Quantity, they are equal between themselves.
Thus, if A=C, and if B-C, then A= B.
4. Things, or Quantities, which are each equal to
half (or any other equal portion) of the same third Thing, are equal to one another.
Thus, if A and B are each equal to half of C, A and B are equal.
$. Things, or Quantities, which are each double, or
triple, &c. of the same third Thing, or Quantity, are equal to one another.
Thus, if D and Ė are each equal twice or three times F, D and E are equal. 6. If to equal Things be added equals, the Sums are equal.
If to the equal Rectangles A and B, be added equals, C and D; then, C +A=D+B.
7. If from equal Things be taken away equals, the remainders will be equal.
If from the equal Rectangles, A and B, be taken away equals, C and D, the remainders of A and B are equal.
3. If to or from unequal Things be added or taken away equals, the Sums or Remainders will be unequal. This necessarily follows from the two laft.
1 9. All Right Angles are equal to one another.
The Right Angle ABC=the Right Angle BDE. 10. Two or more Right Lines, being perpendicular
to the fame Right Line, and being all in the same Plane, are parallel. BC is parallel to DE.
The Angles ABC, BDE, being Right Angles, are equal; by the last.