PROP.XXIV.—Theorem.
PROP.XXIV.—Theorem.
If two magnitudes of the same kind(a,b)have to a third magnitude(c)ratios equal to those which two other magnitudes(a′,b′)have to a third(c′), then the sum(a+b)of the first two has the same ratio to their third(c)which the sum(a′+b′)of the other two magnitudes has to their third(c′).
PROP.XXV.—Theorem.If four magnitudes of the same kind be proportionals, the sum of the greatestand least is greater than the sum of the other two.
PROP.XXV.—Theorem.If four magnitudes of the same kind be proportionals, the sum of the greatestand least is greater than the sum of the other two.
Leta:b::c:d; then, ifabe the greatest,dwill be the least [xiv. anda]. It is required to prove thata+dis greater thanb+c.
Questions for Examination on Book V.
Questions for Examination on Book V.
1.What is the subject-matter of this book?
2.When is one magnitude said to be a multiple of another?
3.What is a submultiple or measure?
4.What are equimultiples?
5.What is the ratio of two commensurable magnitudes?
6.What is meant by the ratio of incommensurable magnitudes?
7.Give an Illustration of the ratio of incommensurables.
8.What are the terms of a ratio called?
9.What is a ratio of greater inequality?
10.What is a ratio of lesser inequality?
11.What is the product of two ratios called?Ans. The ratio compounded of theseratios.
12.What isduplicate ratio?
13.What is Euclid’s definition of duplicate ratio?
14.Give another definition.
15.Define triplicate ratio.
16.What is proportion?Ans.equality of ratios.
17.Give Euclid’s definition of proportion.
18.How many ratios in a proportion?
19.What are the Latin terms in use to denote some of the Propositions of BookV.?
20.When is a line divided harmonically?
21.When a line is divided harmonically, what are corresponding pairs of points called?Ans.Harmonic conjugates.
22.What are reciprocal ratios?
23.Give one enunciation that will include Propositionsxxii.,xxiii.of BookV.
Exercises on Book V.
Exercises on Book V.
Def.I.—A ratio whose antecedent is greater than its consequent is calleda ratio of greaterinequality;and a ratio whose antecedent is less than its consequent,a ratio of lesserinequality.
Def.II.—A right line is said to be cutharmonicallywhen it is divided internally and externallyin any ratios that are equal in magnitude.
1.A ratio of greater inequality is increased by diminishing its terms by the same quantity, anddiminished by increasing its terms by the same quantity.
2.A ratio of lesser inequality is diminished by diminishing its terms by the same quantity, andincreased by increasing its terms by the same quantity.
3.If four magnitudes be proportionals, the sum of the first and second is to their difference asthe sum of the third and fourth is to their difference (componendo et dividendo).
4.If two sets of four magnitudes be proportionals, and if we multiply corresponding termstogether, the products are proportionals.
5.If two sets of four magnitudes be proportionals, and if we divide corresponding terms, thequotients are proportionals.
6.If four magnitudes be proportionals, their squares, cubes, &c., are proportionals.
7.It two proportions have three terms of one respectively equal to three correspondingterms of the other, the remaining term of the first is equal to the remaining term of thesecond.
8.If three magnitudes be continual proportionals, the first is to the third as the square of thedifference between the first and second is to the square of the difference between the second andthird.
9.If a lineAB, cut harmonically inCandD, be bisected inO; proveOC,OB,ODarecontinual proportionals.
10.In the same case, ifO′be the middle point ofCD; proveOO′2=OB2+O′D2.
11.AndAB(AC+AD) = 2AC.AD,or1 AC-+1 AD-=2 AB-.
12.AndCD(AD+BD) = 2AD.BD, or-1- BD+-1- AD=-2- CD.
13.AndAB.CD= 2AD.CB.