Euclid Revised

ADDENDA TO BOOK i.

In addition to the abbreviations already indicated, the following will be used in the Addenda and Exercises.

to signify that the quantity which is placed after it is to be subtracted from that which goes before it.

~to signify that the difference of the two quantities, between which it is placed, is to be taken: this symbol is to be used instead of the one preceding it, when we do not know which of the two quantities is the greater.

The contraction 'sq. on AB' will be still further contracted into AB2. which is to be considered solely as an abbreviation for these words-' the square described on the straight line AB.'

instead of the words 'is not greater than': this symbol includes the possibility of either = or expressing the fact indicated.

Also, for brevity, 'line' means 'straight line.'

Def. A corollary is an obvious inference from a demonstrated proposition.

In the Addenda will be found

1o, all the most evident and important corollaries to the propositions:

2o, some useful deductions, which follow immediately from the propositions, but are not so obvious as to be properly termed corollaries :

3o, some useful theorems, depending only on Book i.

COROLLARIES TO THE PROPS. IN BOOK i.

i. 5. Every equilateral triangle is also equiangular.

i. 6. Every equiangular triangle is also equilateral.

i. 13. (a) If two lines coincide in two separate points they coincide throughout their entire lengths.

(B) If two lines intersect, the sum of the four angles at their common point is equal to four right angles.

(7) All the consecutive angles made by any number of lines drawn from one point, are together equal to four right angles.

(8) If one line meet another, the bisectors of the supplementary angles are at right angles.

i. 16. (a) If one angle of a triangle is right, or obtuse, the other two must each be acute.

(B) Only one perpendicular can be drawn from a point outside a line to the line.

Def. Any line drawn from a point to meet a line, but not perpendicular to it, is called an oblique.

(y) If from a point outside a line there is drawn to the line the perpendicular and any oblique, the foot of the perpendicular will lie on the acuteangled side of the oblique.

(8) In an isosceles triangle the equal angles are acute.

i. 20. Either side of an isosceles triangle is greater than half the base.

i. 29. (a) If two intersecting lines are parallel to two others, the angle between the first pair is either equal or supplementary to the angle between the second pair.

(B) If two angles are equal, and one pair of the sides forming them are parallel, the other pair are also either parallel, or inclined at double the equal angles.

i. 32. (a) Each angle of an equilateral triangle is one-third of two right angles; or two-thirds of one right angle.

(B) If one angle of a triangle is equal to the sum of the other two it is a right angle; and conversely.

(7) If a right-angled triangle is isosceles, each of its acute angles is half a right angle; and conversely.

(8) If two angles of one triangle are equal to two angles of another, the remaining pair of angles are equal.

(e) In a right-angled triangle the acute angles are complementary.

($) In an isosceles triangle each of the two equal angles is half the supplement of the third angle, or the complement of half the third angle. (7) The sum of the four angles of any quadrilateral is equal to four right angles.

i. 33. A quadrilateral which has two sides equal and parallel is a parallelogram.

i. 34. (a) If one angle of a parallelogram is right all its angles are right. (B) If two adjoining sides of a parallelogram (not right-angled) are equal it is a rhombus.

i. 38. (a) A line from a corner of a triangle to the mid point of the opposite side bisects the triangle; and conversely.

(B) If triangles on unequal bases are between the same parallels, then the triangle on the longer base is greater than that on the shorter base; and conversely.

i. 47. (a) A square is half the square on its diagonal.

(B) In a right-angled triangle the square on one of the sides forming the right angle is equal to the difference between the squares on the hypotenuse, and on the other side.

THE FOLLOWING ARE SOME IMMEDIATE DEVELOPMENTS OF THE PROPS. IN BOOK i.-NOT SO OBVIOUS AS TO BE PROPERLY CALLED COROLLARIES.

THEOREM (1)—The difference between two sides of a triangle is less than the third side.

In any ▲ ABC,
AB+ AC > BC,
and AB+BC > AC.

If ACBC, take AC from each side of the first inequality,

and then AB > BC-AC.

If BC AC, take BC from each side of the second inequality,
and then AB > AC-BC.

..always AB > AC ~ BC.

And similarly for the other pairs of sides.

COROLLARIES TO THE PROPS. IN Book i.

i. 5. Every equilateral triangle is also equiangular.

i. 6. Every equiangular triangle is also equilateral.

i. 13. (a) If two lines coincide in two separate points they coincide throu out their entire lengths.

(B) If two lines intersect, the sum of the four angles at their comm point is equal to four right angles.

(7) All the consecutive angles made by any number of lines dra from one point, are together equal to four right angles.

(8) If one line meet another, the bisectors of the supplementary ang are at right angles.

i. 16. (a) If one angle of a triangle is right, or obtuse, the other two mu each be acute.

(B) Only one perpendicular can be drawn from a point outside a lin to the line.

Def. Any line drawn from a point to meet a line, but not perpendicular to it is called an oblique.

(7) If from a point outside a line there is drawn to the line the perpendicular and any oblique, the foot of the perpendicular will lie on the acuteangled side of the oblique.