A BRIEF (Yet full) ACCOUNT OF THE DOCTRINE OF Vulgar and Decimal FRACTIONS. WITH A Specimen of their Demonstrations. By WILLIAM ALINGHAM, Teacher of the Mathematics. LONDON, Printed; Sold by Mr. Mount at the Postern on Tower-Hill, Mr. Lea, Globemaker in Cheapside. Mr. Worgan, Mathematick-Instrument-maker, under St. Dunstan's Church in Fleetstreet; and by the Author, at his House in Channel-Row, Westminster, 1698. THE PREFACE. Reader, THou art here presented with a short Treatise of Vulgar and Decimal Fractions, Quantities of such great and general Use, that without the Knowledge of them there can be no Complete, or Correct Accountant. I have therefore in this Critical Age adventnred to present the Public with this small Tract, in which I have endeavoured to be Methodical and Plain, and to digest the Rules in such Order, with several Exemplications of the same, that I hope they will be rendered intelligible, and serviceable to the meanest Capacity. Lastly, I have added a Specimen of the Demonstration of each particular Operation: All which to the Ingenious I willingly present, hoping of their favourable Acceptance; and that they may receive some Benefit from the same, is the hearty Wish of their real Friend W. Alingham. THE DOCTRINE OF Vulgar Fractions. What a Fraction is, and how read. AN Unite, or Integer, is one whole thing, as, one Pound, one Yard, one Gallon, one Hour, etc. A Fraction, or broken Number, is a part, (or parts) of an Unite, or Integer; and is generally represented by two Numbers set one over the other, with a Line between them, thus, ½. ⅔. ¾. 5/7, etc. The upper number is termed the Numerator; the lower the Denominator, they are read, or pronounced, thus, ½ is one half, ⅔ is two thirds, ¾ is three fourth's, 5/7 is five sevenths, and so of any other, naming the Numerator first, and the Denominator last; the Denominator showing the parts into which the Unite, or Integer, is broke, and the Numerator, the part, (or parts) of the Denominator that is to be taken, or used. Of the Varieties of Fractions. OF Fractions, or broken Numbers, there are four sorts, viz. Proper, Improper, Mixed and Compound. A proper Fraction is, that whose Numerator is lesser than the Denominator, as ½. ⅔. ¾. ⅘. etc. An Improper Fraction, is, that whose Numerator is greater than (or, at least equal to) the Denominator, as 4/3 5/2 6/6 4/4 etc. Mixed, are whole Numbers and Fractions set together thus, 2⅓, 3⅚, 7¾, etc. Compound Fractions are known, by having the word [Of] betwixt them, and are written thus, ⅔ of ¾, also ¼ of ⅚ of ⅞. They are likewise called Fractions of Fractions. Now before we can pass to the Rules of Addition, Substraction, Multiplication and Division in Fractions, they must be prepared, and made fit for such Operation, and this preparation is performed by Reduction, of which there are five kinds, as follows, Reduction the First. TEacheth how to reduce a whole, or mixed Number, into an Improper Fraction, which Fraction shall be equal in value to the said whole, or mixed Number: And contrary, that is, it teacheth to turn an improper Fraction into its Equivalent whole, or mixed Number. CASE I. If it be a whole Number, the Rule is, multiply it by the assigned Denominator, setting the Product thereof for a Numerator over the said Denominator, so shall this Fraction be equal to the given whole Number. Example. Reduce 7 into an Improper Fraction, whose Denominator shall be 4. Note, That these two Lines = is the sign of Equality, as 7 = 28/4 shows that 7 is equal to 28 Fourths. More Examples. Reduce 5 8 9 into a Fraction, whose Denom. is 4 6 7 CASE 2. If a mixed Number is given to be reduced, the Rule is, multiply the whole Number by the Denominator of the Fraction, adding thereto its Numerator, the Sum shall be a new Numerator which, if set over the old Denominator will give a Fraction of the same value with the proposed mixed Number. Exam. Reduce 2¾ into an Improper Fraction. More Examples. Reduce 3 5/7 6 2/3 5 4/9 into an Improper Fraction. CASE 3. If an Improper Fraction is to be reduced into its Equivalent whole, or mixed Number; the Rule is, divide the Numerator by the Denominator, so will the Quotient give a whole Number equal to the Fraction given. Exam. Reduce 28/4 into its Equivalent whole Number. More Examples. Reduce 20/4 48/6 6●/7 into its Equivalent whole Numb. If after dividing any thing remain, set it for a Numerator over the Fractions Denominator, and join the said Fraction to the Quotient. Exam. Reduce 11/4 into its Equivalent mixed Number. More Examples. Reduce 26/7 20/3 49/9 into its Equivalent mixed Numb. This Reduction is absolutely necessary, for there is no working with whole Numbers and Fractions, till the whole Numbers are turned into Fractions. Reduction the Second. TEacheth you how to reduce a com-Fraction to a Simple one, which shall have the same value with it. RULE. Multiply all the Denominators one into another continually, and set the Product thereof for a Denominator; so likewise multiply all the Numerators one into another, and set the Product for a Numerator over the former Denominator, the Fraction thus formed is Equivalent to the given compound Fraction. Example, Reduce ¾ of 8/9 of 9/12 to a simple Fraction. Reduce 4/5 of 5/6 of 8/9 3/6 of 6/7 of 7/9 32/35 of 7/8 of 10/12 to a simple Fraction This Reduction is likewise absolutely necessary, for there is no working with Compound Fractions, and others, till the said Compound are reduced to Simple. Reduction the Third. TEacheth how to abreviate a Fraction, or to find a Number that shall reduce it to its lowest Terms at one Operation, yet still keeping the same value it had at first. RULE 1st. Divide the Numerator and Denominator (if they be both even) by 2.4.6.8. etc. (If the Numerator and Denominator be one even, and the other odd, then try some odd Number, as 3.5.7.9. etc.) that will divide both without a remainder; repeat this Division as often as you can, so shall the last Quotient of the Numerator be a new Numerator, and the last Quote of the Denominator, a new Denominator. Example. Reduce 216/432 into its lowest Terms. More Examples. Reduce 160/270 216/378 2240/3360 into its lowest Terms. But the general way of reducing a Fraction to its lowest Terms, is, to find a Common Measurer, that is, the greatest number, which will divide the Numerator and Denominator without a Remainder, by which means a Fraction is brought to its lowest terms at the first work: For finding of which the Rule is, RULE 2d. Divide the Denominator by the Numerator, and if any thing remain by it, divide the former Divisor, and if after this division any thing remain, divide the last divisor by it: Proceed thus till nothing remain, so shall the last Divisor be the greatest common Measurer, and is a Number that will divide both Numerator and Denominator without a Remainder, and so reduce the Fraction to its lowest terms at one Operation: But, if after all the Divisions are ended there temains one, then is such Fraction in its lowest terms already. Exam. Reduce 216/432 to its lowest Terms. So that 216 is the common Measurer, and is the greatest Number that will divide the Numerator and Denominator without a Remainder. See the work After this method you may try the Examples given, in the first Rule of this Reduction. This Reduction is also very useful, for by it Fractions that are expressed by great Numbers, are made to be expressed by smaller, so that their true value is more easily and readily known. Reduction the Fourth. TEacheth how to bring Fractions of divers denominations into Fractions of one denomination, yet still retaining the same value. RULE. Multiply all the Denominators continually one into another, and set the product thereof for a new Denominator; then multiply the Numerator of the first Fraction into all the Denominators, except its own, the product is the Numerator of the first Fraction, and must be set over the Denominator before found. So likewise for the second Fraction you must multiply its Numerator into all the Denominators, except its own, the product is the Numerator of the second Fraction. Proceed thus with the rest of the Numerators, that is, Multiply each Numerator by all the Denominators, except its own, setting the several Products for new Numerators over the common Denominator first found, so shall these new Fractions be of one denomination, and equivalent to the former. Example. Reduce ⅔ and 4/7 and ⅚ into one Denomination. So that 2/3 = 84/126 4/7 = 72/126 5/6 = 105/126 More Example. Reduce 2/5 & 1/7 & 5/6 3/7 & 1/2 & 5/8 3/2 & ●/5 & 7/9 into one denomination Note 1st. If mixed Numbers are given thus to be reduced, reduce only the fractional parts. Note 2d. If compound Fractions are to be reduced to one Denomination, they must first be brought to simple ones by Reduction the Second. This Reduction is also highly necessary, for before Fractions are brought to the same Denomination, they neither can be Added, nor Substracted. Reduction the Fifth. TEacheth how to alter, or change, a Fraction into another equal in value that shall have any assigned Denominator. RULE. Multiply the Numerator of the Fraction by the assigned Denominator, and divide the Product by the old Denominator, the Quotient is the Numerator to the intended Denominator. Example. Reduce 84/126 into a Fraction, whose Denominator shall be 3. More Examples. Reduce 84/210 48/112 135/360 into a Fraction that shall have 5 7 8 for a Den. Note, If a compound Fraction is thus to be reduced, then by Reduction 2d. turn it to a simple, and then work as the preceding Rule directs. How to find the value of a Fraction. This Reduction is the most useful of all others, for by it the value of any Fraction is found in the known parts of Coin, Weight, Time, etc. And contrary, that is, any part of Coin, Weight, Time, etc. is turned into a Fraction; the method of doing which is as follows, Multiply the Numerator by the parts of the next inferior Denominator, that are equal to an Unite of the same, that the Fraction gives the parts of; the Product divide by the Denominator, the Quote gives the value in the parts you multiplied by: If after this Division any thing remain, multiply it by the next inferior Denomination, dividing the Product by the Denominator, as before. Thus proceed, till you can bring it no lower, so will the several Quotients give the required value of the given Fraction. Example 1st. What's the 6/8 of a Shilling. But if when it be brought to the lowest Terms any thing remain, place it for a Numerator over the former Denominator. Example 2d. What's the 17/19 of a Pound sterling. Example 3d. What's the 5/9 of a Pound weight, Averdupoize. After this method may the value of any Fraction be found (whether it be of Coin, Weight, Time, Liquor Measure, Long Measure, etc.) and given in known and familiar Terms, as in the second Example, where the value of 17/19 of a Pound sterling was required. I answer, that it is 17 Shillings 10 Pence Halfpenny, and 18/19 of a Farthing. So likewise in the third Example, where the 5/9 of a pound weight Averdupoize was required; there I answer, that it was 8 Ounces, 14 Drams, and 2/9 of a Dram. How to turn any part of Coin, Time, Weight, etc. into a Fraction. THis is but the Converse of the former, and therefore (from a little consideration of what foregoes) may be easily effected: For if you do but consider that 1 Shilling is the 1/20 of a Pound sterling, and 1 Penny the 1/12 of a Shilling, and 1 Farthing the ¼ of a Penny: The Names of a Shilling, a Penny, and a Farthing, being only Denominations given them, for the Vulgar in our Nation to know them by, the more universal way of expressing them, being to call a Shilling the 1/20 of a Pound, and a Penny the 1/12 of a Shilling, also a Farthing the ¼ of a Penny: And this way of expressing them (supposing the value of the Pound known) would be intelligable to all Nations that have the knowledge of Numbers; so that if it were required to know what part of a Shilling 9 d. is, I answer, that 'tis 9/12, or when abreviated ¾. In like manner, if it were required to know what part of a pound 4 Shillings, it is evident that 'tis 4/20 or ⅕. But if it were required to know what part of a Pound 1 Penny is, here I must consider that 1 Penny is the 1/12 of 1/20 of a Pound, and therefore if by Reduction the 2d. I reduce it to a simple Fraction, 'twill be 1/240 of a pound Sterling, so a Farthing is ¼ of 1/12 of 1/20 of a pound, and therefore the 1/960 of a pound: Lastly, if it be required to know what part of a Pound 13 s. 5 d. ¼ is, then reducing all into Farthings, it gives 645; and likewise finding the Farthings in one pound which is 960, and setting the former over the later fraction-wise gives 645/960, the part of a pound that 13 s. 5 d. ¼ is. After the same manner may any part of Weight, Time, Measure, etc. be expressed Fraction-wise. Some Examples, with their Answers. What's the 13/14 of a Pound sterling. Answer, 18 s. 6 d. ¾ q. 3/7. What's the 6/7 of a Guinea at 22 s. Answer, 15 s. 8 d. ½ q. 2/7. What's the ⅜ of a Pistol at 17 s. 10 d. Answer, 6 s. 8 d. ¼. What's the 2/9 of a pound Weight Averdupoize. Answer, 3 Oun. 8 Dr. 8/9. What's the 6/7 of a Year, at 365 days. Answer, 312 days, 20 hours, 34 min. 2/7. What part of a Pound sterling is 2 s. 9 d. Answer, 33/249. What part of a pound Averdupoize is 11 Ounces 2 Drams ¼. Answer 713/1024. What part of a Year is 29 days, 14 hours. Answer, 710/8760. The reason of this Reduction is very evident, for in a given Fraction suppose ⅔, such proportion as the Denominator 3 has to its Numerator 2. such proportion has any assigned Denominator, suppose 6, to a Numerator corresponding to it, so that stating the Question according to the Rule of Three, viz. If 3 give 2, what shall 6 give, and then as is directed multiplying the second term 2, which is the Numerator, by the third term 6 the assigned Denominator, dividing the product by the first term 3 the old Denominator, you get 4 for the Quotient, which is a new Numerator to the assigned Denominator. 'Tis by this Reduction we turn a Vulgar Fraction to a Decimal, & contra, also by it all the Decimal Tables are calculated, of which more shall be said when I come to treat of Decimals. Addition of Vulgar Fractions. IF the Fractions to be added have not like Denominators, they must be reduced to a common Denominator by Reduction the 4th. Then add the Numerators together, and set the Sum for a Numerator over the common Denominator, so shall the Fraction thus found be the Sum of the given Fractions. Example. What's the Sum of ⅔ and 5/7. Observe if the Fraction that is the Sum of those two given, happen to be an Improper Fraction, then by Reduction the 1st. Rule the 3d. reduce it into its equivalent whole, or mixed Number as in the last Exam. More Examples. What's the Sum of ⅖ 4/9 What's the Sum of 4/7 6/9 ⅖ Note 1. If you are to add mixed Numbers, add only the fractional parts, being first reduced to a common Denominator by Reduction the 4th. Note 2. If compound Fractions are to be added one to another, or to simple Fractions, than such Compound Fractions must be reduced to simple ones (by Reduction the 2d.) and those again to one Denomination by Reduction the 4th. Note 3. If the Fractions to be added are not parts of the same whole, but the one parts of a Shilling, the other of a Pound, than (by Reduction the 5th.) they must be brought to one Name, or Denomination, i. e. both must be made parts of the same whole. As for Example. What's the Sum of ⅗ of a Pound, and ⅔ of a Shilling. Here they are not only of different demominations, but parts of different Wholes, and therefore is more properly worded thus, What's the Sum of ⅗ and ⅔ of 1/20 of a Pound sterling. Answer, 190/300. Substraction of Vulgar Fractions. THE Rules delivered for reducing and making Fractions fit for Addition, are in all respects and cases to be observed in Substraction; so that whether they are Mixed, Compound, or simple ones, they must be reduced to a common Denominator; then take the Numerator of the Substractor, or Fraction, to be Substracted, from the Numerator of the Substrahend (or Fraction from which we are to Subtract) and set the remainder over the common Denominator, so is this new Fraction the remainder or difference sought. Example. From 1 8/21 or 29/21 subtract ⅔. More Examples. From 38/45 Subtract 4/9. From 201/315 Subtract ⅗. Multiplication of Vulgar Fractions. IF they be simple Fractions to be Multiplied. Then Multiply the two Numerators, together, for a Numerator, and the two Denominators for a Denominator, so shall the Fraction form by these two Numbers be the product Required. Example. Multiply ⅝ by ⅘. More Examples Multiply 7/9; by 8/11. Multiply ⅚ by 2/7. Multiply ⅘ by ¾. Note 1st. If mixed Numbers are to be Multiplied then before you can Multiply them they must be Reduced into Improper Fractions by Reduction the 1st. Rule the 1st. and 2d. Note 2d. If they be Compound Fractions, Reduce them to simple ones. Note 3d. If a whole Number is to be Nultiplyed by a Fraction, then make the whole Number an Improper Fraction by setting one under it. Division of Vulgar Fractions. IF the Fraction to be divided, and also the Fraction by which we divide, that is, Dividend and Divisor, be both simple Fractions, then Multiply the Numerator of the Dividend by the Denominator of the Divisor, and set the product for a Numerator; multiply also the Denominator of the Dividend, by the Numerator of the Divisor, and take the product for a Denominator; the Fraction thus formed is the Quotient. Example. Divide 20/40 by ⅘. More Examples. Divide 56/99 by 7/9. Divide 10/42 by 2/7. Divide 12/20 by ⅘. Note 1st. If either Dividend, Divisor or both, be whole or mixed Numbers, reduce them into Improper Fractions, by Reduction the 1st. Rule 1st. or 2d. and then divide according to the preceding Rule. Note 2d. If they be Compound Fractions reduce them to simple ones by Reduction the 2d. The Rule of Three Direct in Fractions. THE Directions given, both for stating and working Questions in the Rule of Three in whole Numbers, holds also in this of Fractions; so that having framed your Question, as is there directed, 'tis but Multiplying the Fractions in the 2d. and 3d. place together, and divided the product by the first, according to the preceding Rules given for Multiplying and dividing of Fractions, the Quotient is the Answer to the Question. Example. If ⅔ of a Yard of Cloth cost 5/7 of a Pound, what Cost 7/8 of a Yard at that Rate. Proof. More Examples. If 8/9 of a pound Troy cost 5/7 of a Guinea, at 22 s. What shall 9/11 of a pound cost. Answer 405/616. or 14 s. 5 d. ½ 22/77. If ⅘ of a pound Troy cost 3/7 of a Noble, What will 14/15 of a Noble buy. Answer. 392/225 or 1 pound 167/225. If 7/12 of an Hundred weight cost 34 s. 2/14, What will 19 Hundred weight ¾ cost. Answer 453144/392 or 1155 384/392. Note 1. If there be mixed Numbers, reduce them to Improper Fractions. Note 2. If any of the given Fractions be Compound, that is Fractions of Fractions, they must be reduced to simple Fractions by Reduction the 2d. The Rule of Three Reverse in Fractions. HEre also, as in that of whole Numbers, you are to Multiply the second Term by the first, and divide the product by the third, the Quotient answers the Question. Example. If 2/5; of a Yard of Cloth that is a yard broad will make a Garment, How much of 3 Yards wide will make the said Garment. Proof. More Examples. If 54 Men can build a House in 38 days ⅔, How many Men will build the said House in 11 days ⅚. Answer 176 Men 96/213. Lent my Friend ⅝ of a pound for ⅔ of a Year, How much ought he to lend me for 2 Years, to recompense my kindness. Answer 5/24. If when a Bushel of Wheat is sold for 5 shillings ⅔, the penny white Loaf weighs 7 Ounces ¾, What must it weigh when the Bushel of Wheat cost 6 shillings 4/7. Answer, 6 Ounces 377/552. A Collection of pleasant and useful Questions to Exercise the Rules of Vulgar Fractions, By Reduction 5th. WHat's the ⅔ of 17 s. Answer 11 s. 4 d. What's the 5/7 of ⅖ of a Guinea at 21 s. 2 d. Answer, 6 s. 00 d. ½ q. 2/7. What's the ⅔ of half a Mark. Answer 4 s. 5 d. ¼ q. ⅓. What's the ⅗ of a Dollar at 4 s. 2 d. Answer, 2 s. 6 d. What's the 7/9 of 5 pounds. Answer, 3 l. 17 s. 9 d. ¼ q. ⅓. What's the ⅝ of 13 d. ½ Answer 8 d. ¼ q. ¾. What's the 3/7 of 8 Ounces ½ Troy weight. Answer 3 oz. 12 dwt. 20 gr. 4/7. What's the 7/9 of 15 days 3 hours. Answer, 11 days 18 hours 20 minutes. By Reduction 2d. and 5th. What's the ½ of ⅔ of a pistol at 18 s. Answer, 6 s. What's the 5/7 of 8/9 of a Ducat at 7 s. 3 d. Answer 4 s. 7 d. 0 q. 20/21. What's the ¾ of ⅚ of ⅞ of a Guinea at 22 s. 2 d. Answer, 12 s. 1 d. ¼ q. ⅞. What's the ⅜ of ⅕ of 13 l. 4 s. 7 d. Answer 19 s. 10 d. 0 q. ⅛. What's the ⅔ of ¾ of 5 Nobles. Answer 16 s. 8 d. What's the ⅖ of ½ of ¼ of 1 lib. 3 oz. 2 dwt. Troy. Answer, 15 dwt. 2 gr. ⅖. Questions that Exercise most of the preceding Rules. HOw much is ⅔ of ⅚ and ¾ of ½ of a Jacobus at 25 s. Answer 1 l. 3 s. 3 d. 0 q. ⅔. How much is ½ of ⅔ and ⅞ of 9/10. of a Hundred weight Averdupoize. Answer, 1 c. 0 q. 13 lib. 8/15. What Quantity is that, from which if I take 3 5/7 the remainder shall be 1 ⅖. Answer, 5 4/35. What Quantity is that. from which if I take ½ of 5/9 the remainder shall be 2/7 of 5. Answer, 1 89/126. What's the difference betwixt ½, ⅓ and ¼, and a whole Unite. Answer, 1/12. What Quantity is that, to which if I add 3 5/7 the Sum will be 5 4/35. Answer 1 ⅖. What Quantity is that, to which if I add ⅕ of 9/7, the Sum will be 1 89/126. Answer 1681/4410 A person has 2/7 and 1/11 of a Mine, What part is that of the whole. Answer, 29/77. Another Miner has ¾ and 5/7 and 1/16 of a Mine, What share or part is that of the whole. Answer 107/112 A Merchant has 3/16 and ½ of ¼ of a share in the Cargo of a ship, What part is that of the whole. Answer 40/128. or 5/16. Another person has ⅕ of ⅛ and 1/11 of ⅔ of a ship, how much is that of the whole. Answer, 113/1320. Such Questions as the four last, are frequent among those that have parts in Mines, or Ships. What's the product of 3 s. 6 d. by 3 s. 6 d. Here you are to consider that 6 d. is a part of a Shilling, and therefore the Question more rightly proposed is, What's the product of 3 ½ by 3 ½. Answer 49/4 or 12 ¼ that is 12 s. 3 d. Again, What's the product of 3 l. 19 s. 11 d. by 3 l. 19 s. 11 d. Here (as before) consider, that 19 s. 11 d. is 239/240 of a Pound sterling: And so the Question more rightly stated is, What's the product of 3 239/240 by 3 239/240. Answer, 919681/57600 or 15 l. & 55681/57600 of a pound. A certain Person having ⅗ parts of a Coal Mine, sells ¾ of his share for 171 l. What is the whole Coal Mine worth. Answer, 380 l. A Father dying left his Son a certain portion, of which he spent ¼; then of the rest he spent ½, and then he had 252 l. What was the Portion the Father gave him. Answer, 672 l. When the ⅖ of ¾ of a ship is 147 l. 11 s. 3 d. How much is the whole. Answer, 491 l. 17 s. 6 d. A Merchant bought ⅔ of ¾ of a ship, another buys ⅜ of ⅘ of the same, the Question is, Whether their parts were equal, and if not, which had the biggest of the two. Answer, the first Merchant by ⅕. A younger Brother received 210 l. which was ⅜ of ⅔ of his Elder Brother's Portion: Now 3 ½ times his Elder Brother's Portion was 1 ⅓ time his Father's Estate, I demand what his Father's Estate was. Answer, 2205 l. A Person making his Will gave to one Child ⅖ of ¾ of his Estate, and to another ⅚ of ⅔ of his Estate, and when they counted their Portions, the one had 543 l. 1 s. 9 d. more than the other, I demand how much each had, and what was their Father's Estate. Answer, The first had 673 l. 10 s. 9 d. and the second 1180 l. 12 s. 6 d. and their Father's Estate was 2125 l. 2 s. 6 d. A certain Person gave to one of his Children ¾ of ⅖ of his Estate, and of the Remainder he gave another ⅜ of ⅔, and when they told their Money, the one had 173 l. 12 s. 4 d. more than the other, How much had each, and what was their Father's Estate. Answer, the first had 416 l. 13 s. 7 d. ⅕. The Second 243 l. 1 s. 3 d. ⅕. And their Father's Estate was 3388 l. 18 s. 8 d. THE DOCTRINE OF Decimal Fractions. WHAT a Fraction is, and how read, I have already declared in the Doctrine of Vulgar Fractions, and therefore I shall here only show the different way of Noting these from that of Vulgar, with their great use in the Solution of several Arithmetic Questions. A Decimal Fraction is that which hath for its Denominator an Unite, with a certain Number of Ciphers as 10, 100, 1000, 10000, etc. are all Denominators of Decimal Fractions. Hence 'tis evident, that we divide the Unite into 10, 100, 1000, 10000, etc. equal parts. For dividing it first into 10 equal parts, and each of those are again divided into 10 other equal parts; so that the Unite will then be divided into 100 equal parts; and if again we divide each of those hundred equal parts into ten other equal parts, the Unite or Integer will be divided into 1000 equal parts; And so by Decimating the first, and Subdecimating the second, we proceed ad infinitum. Now because all the Denominators of Decimal Fractions differ only in the Number of places, and not in the Figures, they being always an Unite with Ciphers, they may be expressed without their Denominators with a point before them, as 6/10 is thus expressed .6, and 54/100 thus .54, also 27/1000 thus .027 And observe, that this point distinguishes them from whole Numbers. Hence the Denominator of a Decimal Fraction is easily known by the places of the Numerator, the Denominator being always one place more, as .6 hath 10 for its Denominator, and .54 hath 100, and .027 hath 1000 for its Denominator, understand the like of any other. The order of places in Decimals is from left to right, and therefore contrary to the order of places in a whole Number which is from right to left, as in this Decimal .548, here 5 is in the first place next the left hand, and signifies so many tenth parts of an Unite, and is therefore called Primes; the 4 which is in the second place, signifies so many hundred parts of an Unite, and is called Seconds; the figure 8 which possesses the third place from the left hand, denotes so many thousandth parts of an Unite, or Integer, and is called Thirds, and so on, as in the following Table. The Notation Table for Decimals. Primes 3 Tenth parts of Unity. Seconds 5 Hundr. parts Thirds 7 Thou. parts Fourths 9 X Thou. parts Fifths 8 C Thou. parts Sixths 4 Mill. parts This Table consists only of a Decimal Fraction, against which above is set the Value of each place, and below its Name. From a little consideration of what has been said 'tis evident, that Ciphers prefixed on the right hand of the Numerator of a Decimal Fraction, do neither increase nor lessen its value. For .2 is of the same value with .20 or .200 etc. And therefore 'tis very easy reducing Decimal Fractions to a common Denominator, for 'tis but setting Ciphers on the right hand of the Numerator: As suppose .3 and .84 and .476 and .2356 were Decimal Fractions, and it was required to reduce them to one Denominator; here I consider that the Denominator of the greatest Decimal given is 10000, I therefore add so many Ciphers to each of the Numerators that will make each of their Denominators to consist of five places, so that the above proposed Decimals, when reduced, stand thus .3000 and .8400 and .4760 and .2356. I have been as clear as possible in explaining the Notation of these Numbers, because of the great facility they bring with their practice in several Operations, not only in Arithmetic, but in most other parts of the Mathematics. For, had our first Institution of Dividing our Money, Weight, Measure, etc. been Decimally, we had never been troubled with so many Fractions, which cause such great tediousness in several Operations. And indeed the Art of Arithmetic would be taught with much more ease and expedition than now it is, in case such a Reformation should ever be brought to pass. Reduction of Decimals. WHat is here to be done is no more than what was shown in Reduction the 5th. of Vulgar Fractions, only here I shall more largely comment upon what I there but hinted; and show in the first place how to reduce a Vulgar Fraction to a Decimal, and then how to find the Value of any Decimal in the known parts of Coin, Weight, Time, etc. and that with as much brevity and clearness as I can. To Reduce a Vulgar Fraction to a Decimal. THe proportion for reducing a Vulgar Fraction to a Decimal, is, As the Denominator of the Vulgar Fraction to its Numerator, So is 10, 100, 1000, etc. or any assigned Denominator, to its Numerator, that is to the Decimal required. Exam. 1. Suppose it was required to reduce ¾ to a Decimal Fraction, the Operation is as follows, So that .7500 or .75 (for Ciphers on the right hand a Decimal Fraction neither increases nor diminishes its value) is the Decimal equivalent to ¾. Note. From the preceding proportion 'tis evident, That if to the Numerator of any Vulgar Fraction you annex so many Ciphers, as you would have your Decimal to consist of places, and divide by the Denominator, the Quote gives the Decimal required. Exam. 2. Reduce 15/19 to a Decimal of five places. To 15 the Numerator of the given Vulgar Fraction I annex five Ciphers, it makes 1500000, this I divide by 19 the Denominator, the Quote is the Decimal required. See the following Operation. So that the Decimal equal to the given Vulgar Fraction is .78947, which because of the remainder, is not exactly the truth, yet 'tis so near, that it wants not 1/100000 part of an Unite of the truth, and if you proceed farther to make the Decimal consist of six places, it will be .789473, and then it will not want 1/1000000 part of an Unite of the truth; for if the Decimal be made .789474, it would exceed the true value. And thus by Increasing the Number of places in the Decimal you may at last come infinitely near, tho' never to the truth itself. Exam. 3. Reduce 1/32 into a Decimal of five places. Here (because the Decimal is required to five places) I add five Ciphers to 1 the Numerator of the given Fraction, and then divide by 32 the Denominator, the Quote gives 3125 for the Decimal sought. But here Note, that because I annexed five Ciphers to 1 the given Numerator, and there arises but four figures in the Quote, I must supply such defect by prefixing as many Ciphers on the left hand of the first figure in the Quote as there wants places, as in the preceding Example. where the Quote consisted but of four figures or places, here I annex a cipher on the left hand of 3 the first figure in the Quote, and then it becomes .03125 which is the true Decimal required. To Reduce the known parts of Money, Weight, Time, etc. to a Decimal Fraction. FRom what precedes, 'tis evident how the known parts of Money, Weight, Time, etc. may be turned into a Decimal of the same Value, or Infinitely near it, for if in Money, a Pound Sterling be an Integer, whatsoever is less than a Pound, is either a part or parts of the same; and when you know what part or parts thereof it is, you may easily Reduce it to a Decimal of the same Valuc, from what was taught in the last. Exam. What's the Decimal of 9 s. That is Reduce 9/20 into a Decimal consisting of two places. Here working according to what has been before directed, I find the Decimal of 9 s. to be .45 So if I would know the Decimal of 9 d. here I consider that 9 d. is 9/12 of 1/20 of a Pound or 9/240. Working therefore according to the preceding Rule, I find the equivalent Decimal to be .0375. Again, if I would know the Decimal of 3 Farthings, here I must consider that 3 Farthings is the ¾ of 1/12 of 1/20 or 3/960 of a pound, and therefore working as before, I find the Decimal to be .0031 near. Lastly, if it were required to find the Decimal of 7 l. 08 s. ¾ that is 371 Farthings, or 371/900 here repeating the like operation, I find the Equivalent Decimal to be 3864, the like is to be understood in reduceing to Decimals, the known parts of Weight, Time, Measure, Motion, etc. To find the Value of a Decimal Fraction in the known parts of Money, Weight, Time, etc. THis is but the Converse of the former; and therefore the Rule for finding the Value of a Decimal is grounded upon the same reason as that of turning any part of Coin, etc. to a Decimal. For 'twill hold as the Decimal Denominator is to its Numerator, So is the parts in the next Inferior Denomination to the Numerator, or Number of such parts contained in the Decimal. And hence comes this Rule. Multiply the given Decimal by the parts of the next inferior Denomination, that are equal to the Integer the Decimal gives the parts of, and from the product cut of so many figures towards the right hand as there are places in the given Decimal, the remaining figures on the left side are the value of the said Decimal in the next inferior Denomination: If any thing remain, it is the Decimal of an Integer in the Denomination last found, and may be reduced as low as you please by the same Rule, and after the same manner, as it was in Reduct. 5th. of Vulgar Fractions. Exam. How much is .3765 of a pound Sterling. I say, As 10000 to 3765, So is 20 to 7 s. 6 d. See the work. So that from the preceding Work I find the true Value of this Decimal of a Pound sterling .3765 to be 7 s. 6 d. ¼ .44 After a like process may the Value of any Decimal of Weight, Time, Measure, etc. be found. Some more Examples, I might here have added, but I think the Method is so plain, that it will be needless; I shall therefore forbear, and in the room thereof show you a brief, and practical Rule for finding the value of any Decimal of a Pound Sterling, as soon as ever you hear it named, The Rule is The figure in the first place, or place of Primes, being doubled gives you the Number of Shillings, and if the Figure in the second place be 5, or above it, take one Shilling for the five, and add to the former Number of Shillings, found by doubling, then for that which remains above 5, with the Figure in the third place, count so many Farthings less by 1, that those two Figures make, being set in a Numeral Order, or if the Figure in the second place be under 5, then reckon so many Farthings wanting 1, as that and the Figure in the third place of the Decimal make in Number. An Example or two will make it plain. Exam. 1. What's the .375 of a pound Sterling. Here I double 3, which stands in the place of Primes, and that gives 6 s. then because the next Figure (7) is above 5, I add one Shilling to the 6 before found, and it makes 7 s. then the 2 which is left of the 7, with the 5 in the place of thirds, makes 25, which being lessened by 1, giveth 24 Farthings, so that the value of .375 is 7 s. 6 d. Exam. 2. What's the Decimal of .719. Here the first figure 7 doubled gives 14 for the Number of Shillings, as before, and for the other 19 that remains I account 18 Farthings, which is 4 d ½. so that the value of the Decimal .719 is 14 s. 4 d. ½. More Examples might here be given, but I think these are sufficient to illustrate this practical way of finding the valve of the Decimal of a pound Sterling. I shall conclude this with the Insertion of the Decimal Table, for finding the value of any Decimal of a pound Sterling, omitting those of Weight, Measure, Time, etc. because of their being so seldom used, and if required, so easily Calculated from the aforementioned proportion, and likewise for their frequency in Books of this Nature. A TABLE, showing the Decimal of any part of a Pound sterling, & contra. Shillings. 19 .95 18 .9 17 .85 16 .8 15 .75 14 .7 13 .65 12 .6 11 .55 10 .5 9 .45 8 .4 7 .35 6 .3 5 .25 4 .2 3 .15 2 .1 1 .05 Pence. 11 .04584 10 .04166 9 .0375 8 .03333 7 .02917 6 .025 5 .02083 4 .01667 3 .0125 2 .0083 1 .00417 Forth. 3 .00312 2 .00208 1 .00104 The Use of the Table. The method of making this Table is evident from what proceeds, and its use almost as apparent. Let the Decimal of 13 s. 7 d. ½, be required, Seek in the Table first for the Decimal of 13 s. which is .65 next for the Decimal of 7 d. which is .02917, and lastly for the Decimal of ½, which is .00208; I set these Decimals in the order following, and add them together. By which you see the Decimal of 13 s. 7 d. ½ is .68125. In like manner may the Decimal of any other Sum be found, as also the Sum belonging to any given Decimal. Addition of Decimals. AS to the manner of adding, 'tis the the same as in common Addition, the business being only to see that they are rightly placed, according to the manner of their Notation, which thing is easily effected, by setting the point prefixed to them under each other; for then the rest of the places will fall right, whether they be whole Numbers and Decimals, or all Decimals. Some Examples. Here you see in all these cases that Primes stands under Primes, Seconds under Seconds, etc. And where Integers are joined with Decimals, there unites, stands under unites, and Ten under Ten, etc. In which Examples 'tis very plain, that the method of adding is put as it was in whole Numbers, only here you are to make the sum consist of no more Decimal places, then is in the greatest part of it. As in our first Example, the Sum consisted of fix places or Figures, and the greatest part but of five, I therefore cut off five Figures in the Sum, toward the right hand for the Decimal parts, the remainder on the left are Integers. Note, That in this, and the following Rule, the Decimals given to be added, or Substracted, must be parts of the same whole. More Examples. What's the Sume of 29 & 3.007 & .94 & 89.76. Answer. What's the Sume of 3.87 & 486 & .4 & .025. Answer. What's the Sume of 59.4 & 8.796 & 472.6 & .142. Answer. Note, that In. over the preceding and following Sums stands for Integer, and pts. for Parts. Substraction of Decimals. THe Operation here is in all respects like to that in Vulgar Substraction, the main thing (as in the last) being only to see that they are rightly placed, which is done by the direction given in the foregoing Rule of Addition. Some Examples Here you see we Subtract as in common Substraction, only observe, that where the Decimals have not an equal Number of places, the vacances are supplied with cyphers, or are understood so to be, especially in the upper Number. More Examples. From 15 subtract 7.8 Answ. 7.2 From 1 subtract .9872 Answ. .0128 From 58.6 subtract 3.98625. Answ. 54.61375 Multiplication of Decimals. IN Multiplication of Decimals, both the manner of placing and multiplying is in all respects and cases, the same with that of placing and multiplying whole Numbers, the business here being only to find the value of the Product after the Operation is ended, which to do take this general Rule. See how many Decimal places there are in the Multiplicand and Multiplior, and from the Product toward the right hand cut off so many as are in both these, so shall the Figures on the right hand of the point be Decimal places, and those on the left side Integers. But if when the Multiplication is ended, there arise not so many Figures in the Product, as aught to be cut off, then is such defect to be supplied, by annexing as many Ciphers on the Left Hand thereof, as there wants places; with a point before them, and you have the true value of the product; See the following Examples. The consideration of this practice will be of some help to you, in finding the true value of the Quote in Division. More Examples. Of Contraction in Multiplication of mixed Numbers. THere is in this kind of Multiplication, a certain way of Contraction, by which you may get the product, to as few, or many places as you please, without the tedious Multiplication of the whole; the Method of which is as follows. As Suppose 9.58 was to be Multiplied by 8.79, here 'tis evident the decimal will consist of 4 places, and only two would be sufficient. Set down the bigger of the two Quantities for the Multiplicand, and then set the place of Unites in the Multiplior, under that place of parts in the Multiplicand, you would have in the product, and then invert the Order of all the other places in the Multiplior, that is, set the place of Tenns, where Primes should be, and the place of Primes where Tenns should be, and so on with the invertion of the rest; then let each Figure of the Multiplior, Multiply that of the Multiplicand, which is just over, Remembering to add, what would have been brought thither from the following places; then add up all together, and from the Sum cut of two Figures (in this Example) next the Right Hand, and you have your desire, all which by the following Examples, compared with this direction will plainly appear. Example 1st. By the common way By Contraction Example 2d. By the Common way By Contraction This last was required to three places, where you see they are separated by a point toward the Right Hand, being 190, but should have been 192; which small Error is caused by the want of the Carriage from the next row, and therefore if you would have it exactly to 3 places, especially in great Sums, you ought to do it to four. Division of Decimals. THE manner of working Decimal Division, is in every thing like to that of Common Division, and therefore no regard as to their place and Nature is here to be had, any more than what was in Division of whole Numbers; the Mystery of this lying first in their preparation, when need requires. Secondly, In finding the true Value of that Quote after the Division is ended. First, Therefore when it happens, that the Divisor has more places than the Dividend, you must put to the Right Hand of the dividend, (whether it be a whole Number, mixed or Decimal Fraction) a certain Number of cyphers at pleasure, by which it is made fit for Operation. As suppose 14 was to be divided by 361, 'tis evident here is an absolute Necessity of prefixing cyphers to 14 the Dividend, before you can divide by 361 the Divisor. The Dividend being thus prepared take Notice, that there must be as many decimal places in the Divisor and Quotient as are in the Dividend; for the dividend is in effect the product, and the Divisor and Quotient the Multiplicand and Multiplior. And therefore for the finding the value of the Quotient this is the Rule. Look how many Decimal places are in the Dividend, more than in the Divisor, for so many Decimal places will there be in the Quotient. And here Note, that you must be sure to make the Dividend consist of more places than the Divisor, if it doth not so already, which is easily done, by ading cyphers. Example. Having annexed 4 cyphers to each of the Dividends, the first Dividend being an Integer, consists only of the 4 Decimals added; but the later being a Decimal, is made to consist of six Decimals by the 4 cyphers that was added; they being thus prepared, and the work of Division over, you see the Quote consists; of 4 places; now considering how many Decimal places there is in each of those Quotes more than in their proper Dividends, and you shall find that in the first Quote, there ought to be two places of Decimals, and in the second six of the like places, which because there is but four, I prefix two cyphers on the Left Hand thereof, which makes it .004216. Some more Examples. I have to these 4 Examples, set only the Dividends prepared with their Quotients truly Valued, the consideration of which, with the preceding direction, I hope will be a sufficient light in all other Cases that can happen. Of the Use of Decimals. TO show the use of these Numbers in all Solutions, where they might be applied in Expediting an Operation, would be endless, they being of great use in most parts of the Mathematics; but particularly, and principally in Gauging, Surveying, and Measuring, Calculating the Tables of Interest, raising of Logarithmes; as may be seen in most Books, that have Writ of these Subjects, I shall therefore forbear giving Examples of useing them in any of these parts, Except I had Treated distinctly of each of them; and shall close rhis Paragraph with the Collection of a few easy Questions, which are very speedily and easy solved by these Numbers. By Multiplication. IN 756 Pistoles, at 18 s. each, How many Pounds Sterling. Answ. 680 l. 08 s. In 439 Guinea's, at 22 s. 6 d. each, How many Pounds sterling. Answ. 493 l. 17 s. 6 d. If I spend 4 s. 6 d. per Day, How much is that for one Year. Answ. 82 l. 2 s. 6 d. If a yard of Cloth is worth 6 s. 9 d. What comes 59 Yards to at that Rate. Answ. 19 l. 18 s. 3 d. If a piece of Paving be 34 foot, 6 inches long, and 24 foot, 9 inches broad, What's the Content in square feet. Answ. 853 foot .875 If one Man's share in the Cargo of a ship come to 38 l. 14 s. What was the whole worth, supposing there was 158 Men in the ship. Answ. 6114 l. 12 s. If the Interest of 500 l. for one Day is 2 s. 3 d. What's that for a Year. Answ. 41 l. 1 s. 3 d. By Division. IN 680 l. 8 s. How many Pistoles, at 18 s. each. Answ. 756. In 493 l. 17 s. 6 d. How many Guinea's at 22 s. each. Answ. 439. If I spend 82 l. 2 s. 6 d. in one year. What is that for one day. Answ. 4 s. 6 d. If 59 Yards of Cloth cost 19 l. 18 s. 3 d. What Cost one Yard. Answ. 6 s. 9 d. If the Content of a piece of Paving be 853 Foot .875 and the length be 34 foot 6 Inches. What's the true breadth. Answ. 24 Foot 9 Inches. If the whole Cargo of a ship be 6114 l. 12 s. and there be 158 men in the ship. What comes each man's share. Answ. 38 l. 14 s. If the Interest of 500 l. for a Year is 41 l. 1 s. 3 d. What is that for one day. Answ. 2 s. 3 d. I conceive it needless to meddle with the Rule of Three, it being in all kinds and respects performed like that in Vulgar Fractions; I shall therefore leave the Exercise of Questions of this nature to the Ingenious. A Specimen of the Demonstration of the Operations of Vulgar and Decimal Fractions. I Shall first begin with that of Vulgar Fractions, and in order to the more clear apprehending thereof, I must make a short Repetition of what has been already declared in the first page, viz. That a Fraction is a part or parts of some divisible Integer, and is represented by two Numbers, the one above the other beneath a Line thus ⅖. The Number placed beneath the Line is called the Denominator, and shows what parts the Unite is divided into. The Number placed above the Line is called the Numerator, and shows how many of those parts are to be taken in the Fraction. As the Fraction ⅖ denotes two such parts as the Integer contains 5. From this method of expressing Fractions it follows. That every Fraction is to its whole an Unite, as the Numerator is to the Denominator, and consequently. First, That if the Numerator be greater equal less than the Denominator, the Fraction is accordingly greater equal or less than its whole an Unite. The first and second of these kinds are called Improper Fractions, the last are termed Proper. Secondly, That Fractions are not to be estimated by the greatness of their Numbers by which they are expressed, but by the proportion the Numerators bear to the Denominators. Thirdly, That Fractions, whose Numerators to their Denominators bear the same proportion, are equal as ½. 3/6. 10/ 12. Fourthly, That every Fraction is the Quotient of the Numerator divided by the Denominator. This being granted, I propose the following Lemma. Lemma. If a Number multiply two Numbers, their Products are in such proportion to each other, as the Numbers multiplied are to themselves. For 3 multiplying 6 8 produceth 18 24 then I say that 6: 8:: 18: 24. Which thing is evident from the common Notion of Multiplication. For 1: 3:: 6: 18 8: 24 and therefore 6: 18:: 8: 24 by the 11th. of the Fifth of Euclid. But by alternation 6: 8:: 18: 24, which was to be proved. Having laid down this as a foundation, I shall proceed to a Demonstration of each particular Operation. Reduction the First. This is so clear from the nature and manner of expressing a Fraction, as also from the first consequent, that it needs no farther Demonstration. Reduction the Second. The Proof of this is a consequent from that of Multiplication, and therefore I refer it to that place. Reduction the Third. This teaches to abbreviate a Fraction, by dividing both Numerator and Denominator by any Number that will divide both without a remainder. As suppofe 6/8 is given to be reduced to its lowest Terms, here dividing 6 by 2, and 8 by 2, there arises ¾; now since 3 and 4 multiplied by 2 produces the same Numbers, viz. 6/8. therefore 3: 4:: 6: 8 and therefore the Fraction ¾ is equal to 6/8 by the Lemma, and third Consequent. Reduction the Fourth. This teaches to reduce Fractions of divers Denominations into one Denomition, having the same value, for doing of which the Rule is Multiply all the Denominators for a common Denominator, then multiply each Numerator into all the Denominators, except its own, for a new Numerator. As suppose ⅜ and 2/7 were given to be so reduced. From the Operation according to the Rule they will stand thus 3x7/8x7 and 8x2/8x7 that is 21/56 and 16/56, which Fractions are by the aforesaid Lemma and third Consequent equal to those given. Again, Suppose ⅔. ¾. ⅘ were given thus to be reduced. The Fractions that is 40/60 45/60 48/60 are evidently equal to the Fractions proposed by the said Lemma, and third Consequent. Reduction the Fifth. This is proved in the Reduction of a Vulgar Fraction to a Decimal. Of Addition and Substraction of Fractions. THe Fractions whose Sum or Difference is required must be reduced to equal Fractions, having the same Denomination. And then according to the Rule the Sum Difference of the Numerators placed over the common Denominator is the Sum Difference of the Fractions required. Exam. What's the Sum and Difference of ⅜ and 2/7. The Fractions reduced to others equal to them, and of the same Denomination are 21/56 and 16/56 consequently 21 ± 16/56 is the Sum Difference of the Fractions, that is the Sum is 37/56, the Difference 5/56, which was required. Of Multiplication of Fractions. For finding the product of any two Fractions, this is the Rule. Multiply the Numerators together for a new Numerator, and the Denominators together for a new Denominator, the Fraction thus produced is the product. Exam. Suppose ¾ was given to be Multiplied by ⅔, the Product is The reason of which is evident, for by the 4th. Consequent I consider these Fractions as the Quotients of their respective Numerators divided by their Denominators, and so if I multiply ¾ by 2 it gives 2x3/4 for to double any Quotient, is to double the Dividend: But now since I have multiplied by 2, when I ought to have multiplied by ⅓ of 2, therefore ⅓ of this product is the truth, which I effect by tripling the divisor 4. Therefore is the product of ⅔ by ¾. And here you may take notice, that the product of any Quantity multiplied by a Fraction is always less than the Quantity so multiplied. The reason of which follows from the Principle of Common Multiplication, which is, that every Product contains the Dividend so often as the Divisor contains 1 or Unity. If therefore the Multiplior be less than 1 or Unity, the Product will be less than the Multiplicand; For according to this Principle, the Product must not contain the Multiplicand once, because the Multiplior doth not Unity. Hence the Product of two Fractions is evidently less than either of them. And hence also the reason of reducing Compound Fractions to Simple ones is very clear; for to take the ⅔ of ¾, is no more than to Multiply those two Fractions together. Of Division. For working of Division the Rule is Multipliply the Denominator of the Divisor into the Numerator of the Dividend for a new Numerator, and the Numerator of the Divisor into the Denominator of the Dividend for a new Denominator, the Fraction thus form shall be the Quotient. Suppose ⅔ to be divided by ¾. Here, as before, I consider the Fractions as Quotients: So that if the Example had been, How many ¼ is in ⅔; the Answer would be 4x2/3, for that is only Quadrupling the Numerator. But since my Divisor is ¾, 'tis evident the former Quote is 3 times too much; wherefore I take ⅓ of it by tripling the Divifor, and then it stands as above. Of Decimal Fractions. DEcimal Fractions (as I have elsewhere noted) are only Fractions, whose Denominators are an Unite with Ciphers, as 10, 100, 1000, etc. And are conveniently written without their Denominators; for if the Numerators consists of places equal in Number to the Ciphers in the Denominator, then prefix a point before the Numerator, and omit the Denominator; but if the Denominator do not consist of as many places as there are Ciphers in the Denominator, than you must supply that defect by putting Ciphers before the significant figures of such Numerator, with a point on the left hand of such cipher, or Ciphers, As 3/10 78/100 86/1000 54/10000 are thus expressed .3 .78 .086. .0054 To reduce a Vulgar Fraction to a Decimal. SAy, As the Denominator of the Fraction proposed, is to its Numerator, so is 10, 100, 1000 etc. that is, an Unite with as many Ciphers as I intent my Decimal shall have places, to the Numerator of a Decimal equal to it. Example. Reduce ¾ into a Decimal of two places So that 75/100 or .75 is equal to the Fraction proposed by the third Consequent. The 5th. Reduction is here proved, this being the same with that, only then we knew not the name of Decimals, for such Fractions as had 10, 100 etc. for the Denominator. Theorem. The Decimal Fraction .234 is equal to 2/10 + 3/100 + 4/1000 Also the mixed Quantity 3.856 is equal to 8 + 8/10 + 5/100 + 6/1000. For 2/10 = 200/1000 & 3/100 = 30/1000 & 4/1000 = 4/1000 As also 3 = 3000/1000 & 8/10 = 800/1000 & 5/100 = 50/1000 & 6/1000 = 6/1000. But 2/1000+3 1000+3/1000 +4/1000 = the Sum 234/1000 = .234 Also 3000/1000+8 1000+8/1000+5 1000+5/1000+6 1000+6/1000 = the Sum 3. 856/ 1000 = 3.856 by the third Consequent. Hence the first place after the Point in Decimals is the place of Tenths, the second of Hundredths, the third of Thousandths, etc. decreasing in a Subdecuple proportion, from Unity towards the right hand, as whole Numbers increase from Unity towards the left in a Decuple proportion. Hence 'tis easy To Add or Subtract Decimals. The Rule. Place Unites under Unites, Tenths under Tenths, Hundredths under Hundredths, etc. Then add or subtract, as if they were whole Numbers. Exam. in Add. Exam. in Sub. Multiplication of Decimals. MUltiply Integers and Decimals together, as if all were Integers, and then cut off as many places from the Product towards your right hand, or Decimals, as is the Number of Decimal places in both Multiplicand and Multiplior. Example. Here 32.5 is 324/10 and 7.6 is 76/10, which two Fractions multiplied by the Rule given for Multiplication of Vulgar Fractions produce 24624/2100 or 246.24 The same holds in all other. Division of Decimals. DIvide the Dividend by the Divisor in all respects, as in whole Numbers, only observe that so many Decimal places as there are in the Dividend more than there are in the Divisor, so many must be cut off from the Quotient to the right hand for Decimals. Example. The reason of which is evident from Division of Vulgar Fractions. FINIS. Advertisement. ALL Sorts of Mathematic Instruments both for Sea and Land, are most Correctly Made, and Sold, by John Worgan, under St. Dunstan's Church in Fleetstreet.