cond Terms of different Names; (as, for Inftance, in Question the fecond we have this Stating : If 1 142 Feet 1": 26400 Feet) and fo may demand, what Ratio can there be betwixt the firft and fecond: Terms (here Feet and Seconds) and thence conclude that we talk improperly and abfurdly; to which we fhall only answer, that they may imagine the fecond Term to be placed in the third Place, and the third in the fecond Place, as we have hinted in Art. 189, and all Things will be clear; otherwife confider them all as abstract Numbers. Our Reafon for placing them otherwise is only to conform to the common and general Rule, in Art. 187. And it may be observed, that we have put the Word If, (not As) before fuch Statings as have the first and fecond Terms of dif ferent Names, to hint that fuch Stating may be read properly thus, If the firft Number be (give or coft) the second Number, what will the third Number ben (give or coft) and fo the above-mentioned Stating might be read very properly thus: If 1142 Feet give 1 Second, what will 26400 Feet give Polst

206. For the Sake of fuch of our Readers as may have the Curiofity to look into ancient Writers, we'll fhall put an End to this Chapter with the Explanation of fuch Terms as were ufed by the Ancients, in expreffing particular Ratio's; viz, when the Ratio or the Antecedent divided by the Confequent, is Unity, the Ratio is faid to be that of Equality. Multiple Ratio is, when the Antecedent divided by the Confequent is equal to any whole Number; and to exprefs the particular Multiple Ratio's, if the Quo tient was 2, 3, 4, 5, &c. it was refpectively called double, triple, quadruple, quintuple, &c. and fuch are 2 to 1, 3 to 1, 4 to 1, 5 to 1, &c. But the Ratio of a leffer Number to a greater they diftinguished by the Word fub; thus, the contrary to these, or fuch, whofe Antecedent divided by the Confequent is equal to any Fraction, whofe Numerator is an Unit. Whence

*

* In any Fraction, a is called the Numerator, and the De nominator.

Whence the Ratio of 1 to 2, I to 3, 1 to 4, I to 5, &c. or fuch whofe Antecedent divided by the Confequentis,,,, &c. is called, refpectively, fub-duple, fub-triple, fub-quadruple, fub-quintuple,&c.

Super-particular Ratio is, when the Quotient of the Antecedent by the Confequent is an Unit, and a Fraction whofe Numerator is one; and fuch are 3 to 2, 4 to 3,5 to 4, &c. And, to exprefs the feveral Kinds of thefe Ratio's, they write the Word Sefquis before the Name of the leffer Term; thus, the Ratio of 3 to 2 was Sefqui-alteral; 4 to 3, Sefqui-tertian, 5 to 4, Sefqui-quartan, &c. And the contrary to thefe, viz. fuch whofe Quotient of the Antecedent by the Confequent (by fome called the Exponent of the Ratio) is a fractional Number whofe Numerator is greater than Unity; as are thefe Ra-* tio's, 2 to 3, 3 to 4, 4 to 5, &c. are called fub-fuperparticular Ratio's; and thefe particular Ratio's, refpectively, fub-fefquialteral, fub-fefquitertian, fubfefquiquartan, &c.

Super-partient Ratio is, when the Quotient, or Exponent of the Ratio, is an Unit, and a Fraction whofe Numerator is greater than 1; as 5 to 3, 7 to 4, &c. And, to exprefs the particular Kinds of fuperpartient Ratio, they put the Name of the Number by which the Antecedent exceeded the Confequent, betwixt the Words fuper and partient, and the leffer Term of the Ratio's after all; thus, the abovementioned Ratio's were called fuper-bis-partiens tertias, and fuper-tri-partiens-quartas, refpectively, &c. and the contrary to these are called fub-fuper-partient; thus, the Ratio of 3 to 5 was named fub-fuper-bipartiens tertias, &c.

. Multiple-fuperparticular Ratio is, when the Exponent of the Ratio is any Integer greater than an Unit, and a Fraction whofe Numerator is an Unit, as 5 to 2, 10 to 3, &c. and, to exprefs these particular Ratio's, they put the Word Sefqui before the Name of the leffer Term; and before the Word

211. Question 1. If 6 Men could do a Piece of Work in 10 Days, in how many Days could 12 Men do it?

Men Days

Men

Solution. If 6 10 :: 12 reciprocally to the Answer. (See Art. 209.)

X-10

It is evident, that 5 is the true Anfwer; for, if 6 Men could do it in 10 Days, confequently twice 6, or 12 Men, could do it in Half that Time, viz. in 5 Days.

212. Question 2. If, when Wheat is 4 Shillings a Bufhel, the 20 Penny Loaf weighs 18 tb, what ought it to weigh, when Wheat is 6 Shillings per Bufhel? i

Note, The Number 20 in this Question is fuperfluous; for it does not affect the required Price; because we were to find the Price of the fame Quantity, for which Reason it was omitted in the Solution.

Let it be noted once for all, that thefe Questions may be folved by the Rule of direct Proportion; for Example, this Question by Art. 208. may be stated thus,

213. Question 3. How many Yards of Damask of 3 Quarters of a Yard wide muft there be to line a Carpet, that is 12 Yards in Length, and 7 in Breadth?

Yds.

Yds.

Yds.

Solution. As 28 in Breadth: 48 in Length :: 3

Yds.

Yds.

in Breadth.

x by 28

214. Question 4. Suppofe that in a Garrison there are 90 Men, with Meat fufficient for 40 Days; how many Men must be turned out, that the Meat may laft 60 Days?

Days Men Days
90 :: 60 reciprocally.

Men

The Garrifon 90 Men
Meat will ferve 60

Solution. If 40

40

360lo

ᄒ 60

215. As the Learner may be apt to take Things to be in fimple direct Proportion, which are not so, as we have already hinted in Art. 204; fo in this Rule, if he does not reafon with himself, before he ftates the Question, he may take fome Things to be in fimple reciprocal Proportion, which are not fo; for Example, fuppofe that in a Room, where two Men A and B are fitting, there is a Fire; from which A is 3 Feet, and B 6 Feet diftant; and it is required to find, how much hotter it is at A's Seat, than at B's? In folving this Question, at first Sight, the Learner thinking, that as it is evident that, the nearer a Perfon is to the Fire, the greater Heat he must feel, may conclude that this is a Queftion in the Rule of Three reverfe, and therefore to be ftated thus, if K 6 Feet

6 Feet: 1 Degree of Heat: 3 Feet reciprocally: 2 Degrees of Heat; or that the Heat is twice fo great at A's, as it is at B's Seat: But let the Tyro go to a Philofopher, a Perfon who is acquainted with thefe Things, and he will be told, that, according to the Principles of Philofophy, it should be thus ftated, if 6 × 6: 1 :: 3 × 3 reciprocally, or as 3 x 3 : 1 Degree: 6 x 6 directly: 4 Degrees of Heat, or that it is 4 Times fo hot at A's Seat, as at B's. Whence it appears, that, in folving fome Questions which may feem to belong to common Rules of Arithmetic, there is not only required the Knowledge of Arithmetic, but alfo of fome other Science.

216. We fhall put an End to this Chapter, with obferving, once for all, that in the following Part of this Treatife, when we would be understood to mean a reciprocal Proportion, it is always mentioned, and therefore, when a Proportion is not faid to be reciprocal, it is direct.

217.

CHAP. XV.

Of PRACTICE.

Y Practice (gal) is understood fome fhort Methods of folving fuch Questions of of the Rule of Three as are frequent in Business; fo that this Rule might properly go by the Name of Compendiums in the Rule of Three; and, therefore, all the Compendiums in Multiplication and Divifion may be fuppofed to belong to this Rule; but, these being known already, we fhall proceed to lay down a few other Rules adapted to particular Cafes but, firft, the following Table must be committed to Memory.