V. If there be four Numbers in Arithmetical Proportion; the Sum of the first and fourth, is equal to the Sum of the fecond and third. L' Er the four Numbers be in continual Proportion, as these 4, 12, 20, 28, whose common difference is (8.) Again, Let the four Numbers be four difcontinued Proportionals; as thefe, 4, 12, 30, 38. And their common difference (8.) Then In the four Continual Proportions 4, 12, 20, 28 it is evident, That twice the first term 4 (viz. 8.) more by thrice the difference 8 (viz. 24,) is equal to (32,) which is both the sum of the Extreams, (4 and 28,) and of the Means (12 and 20 ) Alfo, In the four discontinued Proportionals 4, 12, 30, 38, it is also as evident, That the first term 4, more by the third term 30, more by once the difference 8, is equal to 42, which is equal to the Sum of the Extreams (4 and 38,) and also of the Means (12 and 32.) VI. If there be four Numbers in Proportion, the Number produced of the first and fourth, bhall be equal to the Number produced of the fecond and third. And if the Number produced of the first and fourth, be equal to that produced of the second and third; Thefe Four Numbers ball be Proportional. ET the four Terms be four Continual Proportionals, as these (2,6, 18, 54.) And let the common Ratio be (3) Then, L 18, Again, Let the four Numbers be four Discontinued Proportionals, as thefe, 2, 6, 54, 162, and the equal Ratio (3.) In the Continual Proportionals it is evident by the Multiplication of Powers: For the fquare of the first term 2 (viz. 4,) multiplied by the Cube of (3) the common Ratio (viz. 27,) the Product will be 108, which is equal to 2 the firft term, multiplied by 54 the fourth term, and also of 18 by 6, the two mean terms multiplied into each other. In the four Discontinued Proportionals 2, 6, 54, 162, it is also as evident. For, 2 (the first term) multiplied into 3 (the common Ratio) produceth (6,) and that into the third term 54, produceth 324; which is equal, both to the Product of 2 into 162 (the two Extreams,) and also of 6 into 54, the two Means. VII. In all Continual Arithmetical Progreffions, the Sum of the Extreams is equal to the Sum of any two of the other Means, taken equidiftant from the Extreams, and to the double of the middle Term; when the Number of Places be Odd. L ET there be these seven Numbers in Continual Arithmetical Progreffion, viz. 3, 6, 9, 12, 15, 18, 21, whofe common difference is 3. Firft, Thefe four Numbers 3, 6, 18, 21, are four Proportionals: Therefore (by the Vth hereof) The Sum of their Means, and the Sum of their Extreams are equal. For the Sum of S3 and 21, the two Extreams 26 and 18, the two Means, are equal to 24. Secondly, Thefe four Numbers 6, 9, 15, 18, are four Proportionals; and therefore, The Sum of their Extreams and Means are equal. For the Sum of 56 and 18, the two Extreams, 29 and 15, the two Means, is equal to 24. Thirdly, These three Numbers 9, 12, 15, are three Proportionals; and therefore (by the fame Vth hereof) The Sum of the Extreams is equal to the double of the Mean. VIII. In Geometrical Proportionals Continual (how many foever the Places be) the Product of the Extreams is equal to the Product of any two of the Mean Terms, equidiftant from the Extreams, and of the Square of the Middle Term; when the Number of Places be Ódd. L ET the Number of Terms be feven, viz. 2, 6, 18, 54, 162, 486, 1458, and their common Ratio 3. Firft, Thefe four Numbers 2, 6, 486, 1458, are four Proportionals; and therefore the Product of the two Extreams fhall be equal to the Product of the two Means: So The Product of S2 2 into 1458, the Extreams, is equal to 2916. 6 into 486, the Means, Secondly, These four Numbers 6, 18, 162, 486, are Proportionals : And (6 into 486, the Extreams,? is is equal to 2916. The Product of 18 into 162, the Means, Thirdly, These three Numbers 18, 54, 162, are three Proportionals; and therefore the Product of the two Extreams, is equal to the Square of the Mean. For Concerning the Rule of Three, or Golden Rule, both Single and Compound. TH HIS Rule may (and most properly) be called the Rule of Proportion; for that it teacheth (in any cafe) by having Three Numbers given, to find a Fourth, which fhall be in Proportion to them; And for the Reason of the manner of Working this Rule, the Theorems delivered in the foregoing Chapter will be Subfervient. Now, this Rule is either Single or Compound. 1. of 1. Of the Single Rule of Three. The Single Rule of Three, is, When Three Numbers are Given, and a Fourth Proportional Number required. And this Rule is either Direct, or Inverfe, or Reciprocal. I. Of Direct. The Single Rule of Three Direct, is, When Three Numbers are given, and a Fourth is demanded, which bears the fame Proportion to the Third, as the Second bears to the First. As in this following Example: If 4 Acres of Ground coft 80 1. What will 8 Acres of the like Ground coft? For the Better understanding of this Operation, look back to the VIth Section of the foregoing Chapter, where you fhall find it demonftrated: That, If there be Four Proportionals, the Product of the First and Fourth, is equal to the Product of the Second and Third. Wherefore, In this Example, If the Product of 8 in 80 (the second and third Terms) produce 640; If that 640 be divided by 4, the first term, the Quotient will be 160; and is the fourth Proportional fought. And for Proof of this, Let the fourth term found (viz 160,) be multiplied by the first term (4,) the Product will be the fame with the Product of the second term (80,) multiplied by the third term (8,) namely (640.) And from hence we may argue thus, If the Product of (80 by 8) be equal to the Product of (4 by the unknown number) Then, The Quotient will be the very fame, whether I divide the Produc of (80 by 8,) or whether I divide the Product of (4 by the unknown number) by 4. For either of the Products being 640, the Quotient muft needs be 160. And now it is manifeft, That if I multiply 4 by a 160, and divide that Product back again by 4, it will give 160 for the Quotient: Because, Whatsoever Multiplication doth, is undone again by Divifion. And this is the true and genuine Reafon of the Operation of the Rule of Three. And that is to say, As (4) is found in (80) just 20 times; fo (8) is found in (160) just 20 times. Or thus, As 4 multiplied by 20 makes (80,) fo (8) multiplied by (20)makes(160.) Another Example. If 80l. will buy 4 Acres of Land, How many Acres will 160 l. buy? If you divide the Product of 4 into 160 (viz. 6.0) by (80,) the Quotient will be (8.) For, as (80) contains (4) 20 times, fo (160) being divided by (20,) the Quotient will be (8) alfo. 2. Inverfe, 2. Inverfe, (or Reciprocal.) The Single Rule of Three Inverfe, is, When there are Three Numbers given, and a Fourth required, which fhall bear the fame Proportion to the Second, as the Third doth to the First. As in this Example. If a certain quantity of Pecks of Oats will keep 8 Horfes for 12 days; Caution. ; The moft Authors that have writ of Arithmetick, have made two diftinct Rules of this Golden Rule,calling the one Direct, and theother Inverse whereas (in truth) they are but Öne: Only care must be taken how to place the Terms given. So in this Example, Look what Proportion 16 Horfes bear to 8 Horfes; the like Proportion do 12 days bear to a fourth number of days: And the Terms ought to be thus placed. As 16 Horfes, to 8 Horfes; fo 12 days to 6 days. And then the Ope ration is the very fame as before: For the Product of 8 by 12 is (96,) which being divided by 16, the Quotient is (6) days. T II. Of the Double, or Compound Rule of Three. HE Double, or Compound Rule of Three, is, When more Numbers I. Direct. Example 1. If 4 Men fpend 19 Pound in 3 Months; How many Pounds will 8 Men fpend in 9 Months? The Solution is thus performed. (1.) If 4 Men spend 19 Pound, what will 8 Men spend? For 19 in 8 (the two Mean Terms) produce 152, which divided by 4 (the given Extream) the Quotient is 38 (the other Extream.) (2.) If 38 Pound be spent by any number of Men in 3 Months; How many Pounds will be spent by the fame number of Men in 9 Months; 23 |