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certain modes of changing either the order or magnitude of proportionals, so as that they continue still to be proportionals.
14. By permutando or alternando ; when it is concluded, that if there be four magnitudes of the same kind proportionals, the first is to the third as the second to the fourth.
Note. In this the antecedents are compared with one another, and the consequents with one another; thus if A: B::C: D, by permutando A:C::B: D. If 1:2:: 3: 6, permutando 1:3 :: 2:6.
This definition differs from those following; in it it is necessary that the quantities should be of the same kind, but in the others it is not. It is called alternate proportion, and is proved by the 35th Prop. of the 5th Book.
15. By invertendo; when it is concluded that if there be four magnitudes proportional, the second is to the first as the fourth to the third.
Note. Here the antecedent is made consequent, and the consequent antecedent. If A: B::C:D then invertendo, B:A::D: C; this is called inverse ratio. The def. is proved by Prop. 20. Book 5.
16. Componendo ; when it is concluded, that if there be four magnitudes proportional, the first together with the second is to the second, as the third together with the fourth is to the fourth.
Note. Hence A+B: B:: C+D:D. Thus if 2 : 1::6: 3, then 2+1:1::6+3 : 3. This def. is proved by Cor. Prop. 21. Book 5.
17. Dividendo; when it is concluded, that if there be four magnitudes proportional, the difference between the first and second is to the second, as the difference between the third and fourth is to the fourth.
Note. If A:B::C:D then AB:B::CD: D. This def. is proved by Cor. 2. Prop. 25. Book 5.
18. Convertendo ; when it is concluded, that if there be four magnitudes proportional, the first is to the sum or difference of the first and second, as the third is to the sum or difference of the third and fourth.
Note. If A: B::C:D, then A: A+B::C:C+D. This def. is proved by Prop. 21 & 25 and by Cor. 1. Prop. 25. Book 5.
19. Ex æquali, or ex æquo; when we infer that if there be any number of magnitudes more than two, and as many others, which taken two by two of each series are in the same ratio, the first is to the last in the first series, as the first to the last in the second series.
Of this there are two species.
20. Ex æquo ordinate. When the first magnitude is to the second in the first series as the first to the second in the second series, and the second to the third in the first series as the second to the third in the second series, and so on, then it is concluded as in the preceding definition that the first is to the last in the first series, as the first to the last in the second. Note.--If there be three magnitudes, A, B, C. And three others, D, E,
Let A:B::D: E,
And B:C::E:F Ex æquo ordinate A:C::D:F This is proved by Prop. 34, Book 5.
21. Ex æquo perturbate ; when the first magnitude is to second in the first series, as the penultimate is to the last in the second series, and the second is to the third in the first series, as the antepenultimate is to the penultimate in the latter series, and so on, then it is concluded as above that the first is to the last in the first series, as the first is to the last in the second. Note. As above, let there be three magnitudes A, B, C,
And three others D, E, F,
Then if A: B::E:F
And B: C::D:E
æquo perturbate A : C::D:F.
END OF DEFINITIONS.
ELEMENTS OF GEOMETRY.
1. SIMILAR rectilineal figures are those which have all their angles respectively equal, and the sides about the equal angles proportional.
2. A parallelogram described on a right line is said to be applied to that line.
3. The altitude of any figure is the right line from the vertex perpendicular to the base.
4. A right line is said to be cut in extreme and mean ratio, when the whole line is to the greater segment as the greater segment is to the less.
5. A right line is said to be cut harmonically when it is divided into three segments, such that the whole line is to one extreme as the other extreme is to the middle part.
Three quantities are said to be harmonically proportional when the 1st: 3d :: difference between the first and second : the difference between the second and third.
Taking any right line b a, (Fig. 1) let it be cut so that bcicd::ba: a d, the three lines b a, ca and da, are in harmonic proportion, since they fulfil the above definition. There is also an exemplification of this in Note 3 to Prop. 3 of this Book.
PROPOSITION I. THEOREM.
Triangles and parallelograms which have the same altitude,
are to one another as their bases.
Part 1. Let the antecedent-base be divided into any number of = parts, and let one of those parts be repeated as often as possible on the consequent base; then connect the extremities of those parts with the common vertex. Since all those small bases
and the As on them of the same altitude, those As are = (38. I. 1.), .. whatever submultiple one of those small bases is of the antecedent base, the same is one of the small as of the antecedent A, and as often as one of those small bases is contained in the consequent base, so often is one of the small As contained in the consequent 4; and in like manner it can be shown that as often as any other submultiple of the antecedent base is contained in its consequent, so often is an equisubmultiple of the antecedent
contained in its consequent; .. the As are to one another as their bases, (Def. 5. 5.).
PART 2. Parallelograms of the same altitude, are to one another as their buses.
Let their diagonals be drawn; then since the As which are the halves of these parallelograms (34. 1.), are of the same altitudes they are to one another as their bases (by part 1);, :. the parallelograms themselves are to one another as their bases.
Cor. 1. Triangles or parallelograms which have equal altitudes are to one another as their bases.
For the bases being placed in directum, the right line joining their vertices shall be parallel to the line in which their bases are; for the perpendiculars from their vertices on the bases are = and par. Then it can be demonstrated as in the Prop., that those As are as their bases.
Cor. 2. Triangles and parallelograms on equal bases are to one another as their altitudes.
If the given As are right angled it is evident, if you consider the given bases as the altitudes, and vice versa. If they are not right angled, construct on their bases right angled As of the same altitudes with them; then those right angled As are as their altitudes; and ., the given As, which are = to them (37. 1), are as their altitudes.
Note 1.-Hence it is evident that if the altitude of an isosceles a be bisected, and right lines be drawn from the point of bisection to the extremities of the base, the whole A will be thus divided into four equal As.
2. If the right line from any < of a A, bisecting the opposite side, be itself bisected ; and if lines be drawn from this point of bisection to the extremities of the bisected side, the whole A will be thus divided into four equal As.
3. If an equilateral a and a right angled isosceles a be on the same base, they are to one another :: ✓3: 1. For the D2 of the altitude of the equilat. = 3 times the O’ of altitude of the right angled A .. the altitudes are to one another :: V3: 1, but the As as their altitudes, .-. &c.
4. If two such As have the same altitude they are to one another ::1:13 For the of the base of the equilateral is of the Ol' of the base of the right angled A, .. those bases are to one another ::1:, and .. the As.
5. Therefore the equilateral A is a mean proportional between those two right angled As; and the two right angled As are to one another :: 1: 3.
PROP. II. THEOR.
If a right line be drawn parallel to any side of a triangle,
it will cut the other two sides, or them produced into proportional segments: and the homologous segments are at the same side of the parallel line.
And if a right line cut two sides of a triangle, or those pro
duced into proportional segments, so that the homologous segments be at the same side of it ; it will be parallel to the remaining side.
PART 1. Connect the extremities of this line drawn par. with the opposite <s; then the As contained by the line drawn par., the connecting lines and the segments of the sides between the par. lines, are = to one another, being on the same base (viz. the line drawn par.) and between the same parallels (viz. this line and the base of given 4,).. each of those = is has the same ratio to the a contained by the line drawn par. and the segts. of the