Quest. 46. If 3481 foldiers are to be placed in a square battle, how many are to be fet in rank or in file? Anfw. 59. (for the fquare root of 3481 is 59.)

36

Quest. 47. If 100l. being put forth for intereft at a certain rate, will at the end of two years be augmented to 112,3% (compound intereft, or intereft upon intereft being computed) what principal and intereft will be due at the first year's end? Anfw. 106. (compofed of tool. principal, and 67. intereft) which 106 is a mean geometrically proportional between 100 and 112,36.

That is 100 x 112,36 = 106 X 106.

Quest. 48. If 100l. being put forth for interest at a certain rate, will at the end of 3 years be augmented to 115,7625% (compound intereft being computed,) what principal and intereft will be due at the first year's end?

Anfw. 105. (compofed of 100l. principal, and 57. intereft) which 105 is the first of two mean proportional numbers between 100 and 115,76257.

Queft. 49. Vitruvius (in lib. 9. cap. 3.) reports, that king Hiero having given orders for the making of a crown of pure gold, was informed that the workman had detained part of the gold; the king being much difpleafed at the deceit, recommended the examination of the bufinefs to the famous Archimedes of Syracufe, who without defacing the crown difcover'd the cheat in this manner, viz. experience telling him that a quantity of gold would poffefs lefs room, or fpace than the fame quantity of filver, and confequently that a mixt mass of gold and filver of the fame quantity would take up fome mean space between the two former; he made a mass of pure gold of the fame weight with the crown, likewise another mafs of filver of the fame weight; then having put the crown, as also the other two maffes feverally into a vefiel filled up to the brim with water, he diligently referv'd the water flowing over into another veffel, and from those three feveral quantites of water fo expelled, he found out the quantity of gold and of filver in the crown: But fince Vitruvius has not deliver'd the practical operation, I fhall fhew the fame after the manner of Cardanus, Gemma Frifius, and other Arithmeticians.

Let us therefore suppose the weight of the crown, as also of the two several maffes, to have been 56. Suppose also that by putting the mafs of gold into the veffel, 3lb. of water was expelled; by putting in the crown 31lb. and by putting in the mafs of filver 416. The question therefore is to know how

much gold and how much filver the crown was compofed of. This may be refolved by the Rule of False, after this manner : Suppofe 3. of gold to be in the crown, then there remained 2 lb. of filver; now fay by the Rule of Three, if 5lb. of gold expel 3lb. of water, how much 3lb. of gold? Anfw. 1 lb. Alfo if 5 lb. of filver expel 4 of water, how much 2 lb. of filver? Anfw. 1lb. of water; add therefore the water of the filver and of the gold together, to wit, I and I, so there will arife 33 lb. of water: This ought to have been 3:16. (for fo much overflow'd by putting in the crown ;) but it is too much by ; therefore is to be noted with+, for the error of the firft pofition 3lb. Again, feign another quantity of gold to have been in the crown, to wit, 2lb. therefore there remain'd 3lb. of filver; then fay, if 5 lb. of gold expel 3lb. of water, how much 2 lb. of gold? Anfw. 1 lb. of water: Alfo if 5lb. of filver expel 4lb. of water, how much 3lb. of filver? Anfw. 2,7%, then add 1lb. unto 2%, the fum will be 3% of water. This ought to have been 3lb. but it is too much by 13. be noted with + for the error of the fecond pofition 2 lb. Here because the errors are Fractions having a common Denominator, I take their Numerators, 7 and 13 instead of the errors, then multiplying crofs-wife, to wit, 3 by 13 the product is 39; alfo 2 by 7, the product is 14, which fubtracted from the former product 39, (because the errofs are alike,) leaves 25 for a Dividend; alfo the difference between the errors 7 and 13 is 6 for a Divifor: Laftly, dividing 25 by 6 the quotient is 4; fo much gold, therefore,

39

3

7

Therefore

25

6

48

is to

14 2

13

was in the crown, and confequently (because the weight of the crown was 5lb.) there was of filver, which may be proved thus: Say, if 5lb. of gold expel 3lb. of water, how much 4 lb. of gold? Anfw. 24lb. of water: Again, if 5lb. of filver expel 4 of water, how much of filver? Antw. b. of water, which being added to 21 lb. the fum is 3 lb. of water, to wit, as much as flowed over when the crown was put into the veffel.

Here note, that in making a trial of this nature, there is no neceffity that the mafs of gold or of filver be of the fame weight with the crown, or whatsoever thing is to be examined, but of what notable part of the weight you please.

3

CHAP.

A

CHA P. XLIV.

Of SPORTS and PASTIMES.

741. PROBL. I.

To difcover a number which any one fhall have in his mind, without requiring him to reveal any part of that or any number whatsoever.

A

FTER any one has thought upon a number at pleasure, bid him double it, and to that double bid him add any fuch even number as you please to affign; then from the fum of that addition let him reject one half, and referve the other half: Laftly, from this half bid him fubtract the number which he first thought upon; then may you boldly tell him what number remains in his mind after that fubtraction is made, for it will always be half the number which you affigned him to add.

For Example: Suppofe he thought upon 6, the double thereof is 12, to which bid him add fome even number at your pleasure, fuppofe 4, fo will the fum be 16, whereof the half is 8, from which if he subtract 6, (the number first thought on) the remainder is 2, (to wit, half the number 4, which was by you affigned to be added ;) which remainder you discover, notwithftanding all the operation was performed in his mind, without his making known any number whatsoever. Note, That the adding of an even number, as aforefaid, is not of neceffity, but only to avoid a fraction that will arife, by taking the half of an odd number.

742. The Reason of the Rule.

If to the double of any number (which number for diftinction fake I call the firft) a fecond number be added, the half of the fum muft neceffarily confift of the faid firft number, and half the fecond; therefore if from the faid half fum the firft number be fubtracted, the remainder muft of neceffity be half the fecond number which was added,

743 PROBL. II.

Two numbers, the one even and the other odd, being proposed unto two perfons, to the end they may (out of your fight) feverally chufe one of thofe numbers; to discover which of these numbers each perfon had chofen.

Suppose you have propounded to Peter and John two numbers, the one even and the other odd, as 10 and 9, and that each of those perfons is to chufe one of the faid numbers unknown to you. Now to discover which number each person made choice of, you must take two numbers, the one even and the other odd, as 2 and 3; then bid Peter multiply that number which he has chofen by 2; and caufe John to multiply that number which he has pitched upon by 3; that done, bid them add the two products together, and let them make known the fum to you, or else demand of them whether the said fum be even or odd, or by any other way more fecret endeavour to difcover it, by bidding them take the half of the faid fum, for by knowing whether the faid fum be even or odd, you obtain the principal end to be aimed at; because if the said fum be an even number, then infallibly he that multiplied his number by your odd number, (to wit, by 3) did chufe the even number (to wit, 10;) but if the faid fum happen to be an odd number, then he whom you caufed to multiply his number by your odd number, (to wit 3) did infallibly chuse the odd number (to wit, 9.)

For Example: If Peter had made choice of 10, and John 9, fuppofe you required Peter to multiply his number 10 by 2, and John to multiply his number 9 by 3; the products will be 20 and 27, whereof the fum is 47, which being an odd number, you may thence conclude that John, whom you caused to multiply his number by 3, chose the odd number 9, and therefore Peter took 10. But if you had ordered John to multiply his number 9 by 2, and Peter to multiply his number 10 by 3, the products would have been 18 and 30, whereof the fum is 48, which is an even number; from whence you may infer, that he that multiplied his number by 3 pitched upon the even number, and therefore Peter chofe 10, and John 9.

The reafon of the faid Rule will appear from Note 1. and 2. to Art. 64. and Note 1. and 3. to Art 47.

744. PROB L. III.

A certain number of diftinct things being propounded, to difpofe them in fuch an order, that cafting away always the ninth, or the tenth, or any other that shall be affigned, to a certain number, thofe remaining may be fuch as were first intended to be left.

This Problem is usually proposed in this manner, viz. Fifteen Christians and fifteen Turks being at fea in one and the fame fhip in a terrible ftorm, and the pilot declaring a neceffity of cafting the one half of those persons into the fea, that the rest might be faved; they all agreed, that the perfons to be caft away fhould be fet out by lot after this manner, viz. the thirty perfons fhould be placed in a round form like a ring, and then beginning to count at one of the paffengers, and proceeding circularly, every ninth perfon fhould be caft into the fea, until of the thirty perfons there remained only fifteen. The question is, how thofe thirty perfons ought to be placed, that the lot might infallibly fall upon the fifteen Turks, and not upon any of the fifteen Chriftians? For the more eafy remembring of the Rule to refolve this question, 'I fhall prefuppofe the five vowels, a, e, i, o, u, to fignify five numbers, to wit, (a) one, (e) two, (i) three, (o) four, and (u) five; then will the Rule itSelf be briefly comprehended in these two following verfes:

From Numbers, Aid, and Art,
Never will Fame depart.

In which verfes you are principally to obferve the vowels, with their correspondent numbers before affigned; and then beginning with the Chriftians, the vowel o (in from) fignifies, that four Chriftians are to be placed together; next unto them the vowel u (in Num.) imports that five Turks are to be placed together; in like manner e (in bers) denotes two Chriftians; a (in Aid) one Turk; i (in Aid) three Chriftians; a (in and) one Turk; a (in Art) one Chriftian; e (in ne) two Turks; e (in ver) two Chriftians; i (in will) three Turks; a (in Fame) one Chriftian; e (in Fame) two Turks; e (in de) two Chriflians; a (in part) one Turk.

745. The invention of the faid Rule, and fuch like, depends upon the fubfequent procefs, viz. If the number of perfons be thirty, let thirty figures or cyphers be placed circularly, or elfe in a right line as you fee,