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to the samo proportion lo any part of their sum, &s the whole compotind is 10 any part of the compound, which cractly accords with the principles of Fellowship Honce we have the following
RULE. As the sum of the PROPORTIONAL QUANTITIES found by linking as before : is to EACH PROPORTICNAL QUANTITY ; : so is the WHOLE QUANTITY or compound requircd : 10 the REQUIRED QUANTITY of cach. \Ve will now apply this rule io pxerforming the Innt question. 4 3
10:3:: 240. 72 lbs., at 4 cents. S
10:1: : 240: 24 los., at 6 cents. 9
Ans 10:2:: 240 : 48 lbs., at 9 vents. 11.
10: 4:: 240 : 96 103, at licento.
10 1. A grocer, having sugars at 8 cents, 12 cenca, and it conts per pound wishes to make a composition of 120 lbs., worth 13 cents per posud, withon gain or loss; what quantity of cach must be taken?
A. 30 lbs. at 8, 30 lbs at 12. and it'15: w16. 2. How much water, at 0 per gallon, must be mixed with wine, at du centa per gallon, so as to fill a vessel of 90 gallons, which may be offered at 50 cents per gallon? A. 56 gallons of wine, and 33ę gallons of water.
3 How much gold, of 15, 17, 18, and 2 carats fine, must be mixed together, to form a composition of 40 ounces of 20 carats fine?
A. 5 oz. of 15, of 17, of 18, and 25 oz. of D.
INVOLUTION. TLXXXIV. Q. How much does 2, multiplied into itself, or by 2, make
Q. How much does 2, multiplied into itself, or by 2, and that prodoct by 2, Q When a number is multiplied into itself once or more, in this mannut, what is the process called ? A. Involution, or the Raising of Powers.
Q. What is the number, before it is multiplied into itsolf, called? A. The first power, or root.
Q. What are the several products called ? A. Powers. d. In multiplying, 6 by 6, that is, 6 into itself, making 36, we use 6 twice what, then, is 36 called? A. The second power, or square of 6.
Q. What is the 2d power or square of 8?, 10? 12? A. 64, 100, 144. d. In multiplying 3 ly 3, making 9, and the 9 also by 3, making 27, we uso ho thrce 3 times ; what, then, is thc 27 called? A. The 31 power, or cube of 3. Q. What is the 3d power of 2? 3? 4? A. 8, 27, 64.
Q. What is the figure, or number, called which denotes the power, as, 3d power, 2d power, &c. ? A. The index or exponent.
Q When it is required, for instance, to find the 3d power of 3, what is the index, and what is the power? A. 3 is the indes, ?i the power.
Q. This index is sometimes written over the number to be multiplied, thus, ; what, then, is the power denoted by 2'? A. 2X2X2X2= 16.
Q. When a figure has a small one at the right of it, thus, 65, what does it meant A. The 5th power of K, of that 5 mugć bo raised to the 5th power.
1. flow much is 122, or the square of 12? A. 144 9 How much is f?, or the squarc of 4? A 16 3 How much is 102, or the squarc of 107 A. 100 4. How much is 43, or the cube of 47 A 64. 5. How much is 1“, or the 4th power of 1 ? A. I 6. What is tho biquadrate or 4th power of 3 A 81 7. What is the squarc of 1? ? A. 1, $. 8. What is the cube of 1? ? ? A. $ , of 9. What is the square of ,5 1,2? A. ,25; 1,44. 3. Involve 2 to the 20 power; 2 to the 3d pc ker. A. 4, 8 11 Involve f to the 2d power ; Po to the 21 nower. A.
Tåg 12 Involve of to the 20 power. A tem=* 13 Involve to the 2d power. 14. What is o?, or the square off? A. H. 15. What is the value of 2'? A. zs. 16. What is the value of t'? A. er:
Exercises for the Slate. 1 What is the square of 900 ? 810000. 2. What is the cube of 211? 1. 9393931. 3. What is the biqundrate or 4th power of 80 ? A. 40960000. 4 What is the sursolid or the 5th power of 7? A. 16807. 5. Involve 12, s, ig, cach to the 34 power. A 1931, 33 5091 : 6. What is the square of 51. Å 30.. 7. What is the square of 1.}? A. 2724. & What is the value of 8" ? A. 32763. 9. What is the value of 104? A. 10000 10. What is the value of 64 ? A. 1296. 11. What is the cube of 25 ? A. 15625.
EVOLUTION. LXXXV. Q. What number, multiplied into itself, will make 16. hat is, what is the first power or root of 16 ? A. 4.
Q. Why? A. Pecuuse 4 X 4= 16. Q. What number, multiplied into itself three times, will make 27?that is what is the 1st power or root of 273 ? A. 3.
Q. Why? A. Because 3x3x3= 27.
Q. What, then, is the metliod of finding the first powers or roots of 21 jd, &c. powers called?' A Evolution, or the Ezeraction of Roots.
He la Involution wo wore required, with the first power or root being given, to und higher powers, as 2d, 3d, &c. powers; but now it seems, that, with the 241, 3il, &c. powers leing given, we are required to find thio lst power or root again ; how, then, does Involution differ from Evolution ? A. It is ezactly the opposite of Involution.
Q. How, then, may Evolution be defined ? A. It is tho method of finding the root of any number.
1. What is the square root of 144? d. 12. Q. Why? A. Bocause 12 X 12= 141. 2. What is the cube root of 27 ? A. 3. Q. Why? A. Because 3 X3 X3=27. 3. What is tire biquaclrate root of 81 ? A. 3. Q: Why? A. Because 3 X3X3X3=81. We have seen, that any number may be raised to a perfect power by Involution; but there aro many numbers of which precise roots cannot bo ob tained ; as, for instance, the square root of 3 cannot be exactly determined, there being no nuinber, which, by being multiplied into itself, will make 3. By the uid of decimals, however, we can come nonrer and noarer; that is, approximate towards the root, to any assigned dogrco of exactness. Those numbers, whose roots cannot exactly be determined, are called SưRD Roots, and thoso, whose roots can exactly he determined, aro called RATIONAL Roots.
To show that the square root of a nuinber is to be extracted, we prefix this character, V. Other routs are denotod by the same character with the index of the required root placed before it. Thus VI significs that the square roct of 9 is to be extracted ; 'V27 signifies that the cube root of 27 is to bo extracted
j V64 = the 4th root of 64.
When we wish to express the powor of scveral numbers that are connected 'ogether by these signs, +2.-, &c., a vinculum or parenthesis is used, drawn From the top of tho sign olího root, and extending to all the parts of it'; thus cho cube root of 30 - 3 is expressed thus, 30 — 3, &c.
EXTRACTION OF THE SQUARE ROOT.
Q. We have ecen (TT LXXXV.) that the root of any nuniber is its 1st power; also that a square is the 2d power : what, then, is to be done, in order to find the lst power; that is, to extract the squaro root of any number?
A. It is only te firul tha! number, which, being multiplied into itself, roill preduce the given number.
Q. We have seen ( LXXIX.) that the process of finding the contents of a square consists in multiplying the length of one side into itself; when, shen, the contents of a square are given, now can we find the length of each side; or, to illustrate it by an examplc, !f tho contents of a square figure bo 9 feet, what must be the length of cach side ? A. 3 feet.
Q. Why? A. Because 3 it. X 3 it. = 9 square feet.
Q. What, then, is the difference in contents between a square figure whose sides are each 9 féct in length, and one which contains only 9 square feet?
9X9=&! -9=72, Ans R. What is the difference in contents between a square figure containing 3 square feet, and ono whose sides are each 3 feet in length ? A. 6 square feet
Q. What is the square root of 144 ? or what is the length of each side of a figure, which contains 144 square feet? A 12 square feet. Q. Why? A. Because 12 X 12,144
@ How, then, may we know if the root or answer be right? A. By mult plying the root into itself ; if it produces the given number, it is right.
Q. If a square garden contaiiis 16 square rods, how many rods does it meas urt or each side? and why?
A. 4 rods. Because 4 rods x 4 rods = 16 squarc rods 1. What is the square fout of 64! and why? 2. What is the square root of 100? and why? 3. What is tho square root of 49? and why? 4. Extract the square root of 144. 5. Extract the squaro rooi of 36. 6 What is the square root of 3600 ? 7. What is the square root of ,25 ? A. ,5. 8. What is the square root of 1,44? A.'1,2. 9 What is the value of V25! or, what is the squarb root of 25) 10 What is the value of V,1?. 1. ,2. II Wnit is the square root of 1 ? A. f. 12. What is the value of $? A. 13. What is the square root of foff? A. H. 14. What is the square root or 64? Vot=1/9 = 5= 24, Ans 15. What is the value of Vf of? 16. What is the square root of 301?
17 What is the differenco between the square root of 4 and the square of 1. which is the same thing, what is the differenne bet woen V4 and 4">
V4=2, and 4? = 16 ; then, 16 - 2=14, ANS What is tho difference between V9 and 92 ? 19 What is the difference between V 16 and 19? 20. What is the difference between Vot and ? A. O.
2!. There is a square room, which is calculated is accomodate 100 schol. 18; how many can sit on one sido ? :
21. 18400 boys, having collected together to perform some military evolutions, should wisk to march through the town in a solid phalaox, or square body, of how many must i!ic first rank consist ?
2?. ) general han 400 men ; how many must he place in rank and file to furn them into a square ?
24. I certain suure pavement contains 1500 square stones, all of the sanie *70 ; 1 femand how many are contained in one of its siiles ? A. 40.
25. A man is desirous of inaking his kitchen garden, containing 24 acres, or 499 rols, a contplete square; what will be the length of one side?
26. A'squaro lot of land is to contain 224 acres, or 3600 roxls of ground bril, for the sako of fruit, there is to be a smaller suare within the larger, which is to contain 223 rods : what is the length of each side of both stros?
A. 60 rods tho outer, 15 rods the ir.vet.
Exercises for the Slate. 1. If a square field contains 6400 square rols, how many rods in length oes I Weitspre on each si:le? A. 80 rods.
2. How many trees in each row of a square orcharl, which contains !! rees? A. 500 tr 'es
3. A general has a brigada consistmg of 10 regiments, cach regiment of 1 companies, and each company of 100 men: how many must be placed in raus will, to form them in a coinnlete square? A.'100'men.
4. What is the square root of 250W? A. 50.
7. What is the difference between the aquare root of 36. and the square of 38?
A. 1290 8. What is the difference between V4900 and 4900* ? A. 24009930. 9. What is the differenco between V81 and 81" ? A. 6552. 10. Wha:. is the difference between Vf and ? Viso = $, and *=h; then, 7-1386 1296
=ff, Ans 1 What is the difference between V Po and > A. 12. Wlzat is the amount of V4 and V9 ? A. 5. 13. What is the sum of V4 and 9% ? A. 83. 14. What is the amount of V301 and V2724? A. H.
15. What is the length of one side of a squuro garden, which contains 1296 Muaro rods? in other words, what is the square root of 1906 ?
In this example we have a little difficulty in ascertaining the root. This, erhups, may be obviated by oxamining the following figure, (which is in the irn of the garden, and supposed to contain 1296 squaro rods,) and carefully wuting down the operation us we proceed. OPERATIONS.
In this examplo we know that 1st.
2d. the root, or the lorigth of one side square rods. square rods.
of tho garden, must be greater 30)1296(30
than 30, for 30% = 900, and less 900
than 40, for 40% = 1600, which is
greater than 1296; therefore, we 80 + 6=683396(8 66)396
take 30, the less, and, for correo 396
nience'sake, write it at the left of 1290; as a kind of divisor, like
wise at the right of 1296, in the 0
form of a quotient in division ;
(See Operation 1st.); then, sub Fig.
racting the square of 30, = 900 30 rods.
sq. rods, from 1296 sq. rods, leaves 6 rods.
396 39. rods. 20
The pupil will bear in mind, B
that the Fig. on the left is in the 180
form of the garden, and contains the same number of square rods. viz.. 1.296. This figure is divided into
parts, called A, B, C, and D
It will be perceived, that the 906 A
square rous, which we deducted, ure found by multiplying the
length of A, being 30 rods, Ty she rods.
breadth, being also 30 rods, that 30, length of A.
30 is, 30% =900, 30, breadth of A.
'16 obtuin the square rous in 900, sq. rods in A.
B, C, am! D, the remaining parts 180
of the figure, we may multiply the length of euch by the breadth of each, thus; 30 6 = 190,
6x6 = 36, and 30X6= 180; 30 rods
6 rods. then, 180 + 36 + 180 = 396 of B, that is, 30, to the length of D, which is also 30, making GO; or, which is
square rods : or, add the length the same thing, we may double 30, making 60 ; to this add the length of C, 6 rods, and the sum is 66.-Now, to obtain the square rods in the whole lengib of B, C, and I, we multiply their length, 6 rods, by the breadth of each sida, thus ; 66 6 396 square rods, the same us before.