Book 2 — APPENDIX TO (continued 2)

This last amounts to the solution of the following: In how many ways can n counters (undistinguishable from each other) be distributed into m boxes? And the following will now be easily proved: The number of ways of distributing c undistinguishable counters into b boxes is (b - 1)_{b + c - 1}, if any box or boxes may be left empty. But if there must be 1 at least in each box, the number of ways is (b - 1)_{c - 1}; if there must be 2 at least in each box, it is (b - 1)_{c- b-1}; if there must be 3 at least in each box, it is (b - 1)_{c - 2b - 1}; and so on.

The number of ways in which m odd numbers can be put together to make n, is the same as the number of ways in which m even numbers (0 included) can be put together to make n-m; and this is the number of ways in which m numbers (odd or even, 0 included) can be put together to make ½(n-m). Accordingly, the number of ways in which m odd numbers can be put together to make n is the same as the number of combinations of m-1 things out of ½(n-m) + m-1, or ½(n + m)-1. Unless n and m be both even or both odd, the problem is evidently impossible.

There are curious and useful relations existing between numbers of combinations, some of which may readily be exhibited, under the simple expression of mₙ to stand for the number of ways in which m things may be taken out of n. Suppose we have to take 5 out of 12: Let the 12 things be marked a, b, c, &c. and set apart one of them, a. Every collection of 5 out of the 12 either does or does not include a. The number of the latter sort must be 5₁₁; the number of the former sort must be 4₁₁, since it is the number of ways in which the other four can be chosen out of all but a. Consequently, 5₁₂ must be 5₁₁ + 4₁₁, and thus we prove in every case,

mₙ′ = mₙ₋₁ + (m - 1)ₙ₋₁

0ₙ and nₙ both are 1; for there is but one way of taking none, and but one way of taking all. And again mₙ and (n-m)ₙ are the same things. And if m be greater than n, mₙ is 0; for there are no ways of doing it. We make one of our preceding results more symmetrical if we write it thus,

2ⁿ = 0ₙ + 1ₙ + 2ₙ + ... + nₙ

If we now write down the table of symbols in which the (m + 1)th

0 1 2 3 &c.

1 0₁ 1₁ 2₁ 3₁, &c.

2 0₂ 1₂ 2₂ 3₂, &c.

3 0₃ 1₃ 2₃ 3₃, &c.

&c. &c. &c. &c. &c.

number of the nth row represents mₙ, the number of combinations of m out of n, we see it proved above that the law of formation of this table is as follows: Each number is to be the sum of the number above it and the number preceding the number above it. Now, the first row must be 1, 1, 0, 0, 0, &c. and the first column must be 1, 1, 1, 1, &c. so that we have a table of the following kind, which may be carried as far as we please:

0 1 2 3 4 5 6 7 8 9 10

1 1 1 0 0 0 0 0 0 0 0 0

2 1 2 1 0 0 0 0 0 0 0 0

3 1 3 3 1 0 0 0 0 0 0 0

4 1 4 6 4 1 0 0 0 0 0 0

5 1 5 10 10 5 1 0 0 0 0 0

6 1 6 15 20 15 6 1 0 0 0 0

7 1 7 21 35 35 21 7 1 0 0 0

8 1 8 28 56 70 56 28 8 1 0 0

9 1 9 36 84 126 126 84 36 9 1 0

10 1 10 45 120 210 252 210 120 45 10 1

Thus, in the row 9, under the column headed 4, we see 126, which is 9 × 8 × 7 × 6 ÷ (1 × 2 × 3 × 4), the number of ways in which 4 can be chosen out of 9, which we represent by 4-{9}.

If we add the several rows, we have 1 + 1 or 2, 1 + 2 + 1 or 2², next 1 + 3 + 3 + 1 or 2³, &c. which verify a theorem already announced; and the law of formation shews us that the several columns are formed thus:

1 1 1 2 1 1 3 3 1

1 1 1 2 1 1 3 3 1

1 2 1 1 3 3 1 1 4 6 4 1 , &c.

so that the sum in each row must be double of the sum in the preceding. But we can carry the consequences of this mode of formation further. If we make the powers of 1 + x by actual algebraical multiplication, we see that the process makes the same oblique addition in the formation of the numerical multipliers of the powers of x.

1 + x

1 + x

1 + x

x + x²

1 + 2x + x²

1 + 2x + x²

1 + x

1 + 2x + x²

x + 2x² + x³

1 + 3x + 3x² + x³

Here are the second and third powers of 1 + x: the fourth, we can tell beforehand from the table, must be 1 + 4x + 6x² + 4x³ + x⁴; and so on. Hence we have

(1 + x)ⁿ = 0ₙ + 1ₙx + 2ₙx² + 3ₙx³ + ... + nₙxⁿ

which is usually written with the symbols 0ₙ, 1ₙ, &c. at length, thus,

(1 + x)ⁿ = 1 + nx + n n - 1 x² + n n - 1 n - 2 x³ + &c.

2 2 3

This is the simplest case of what in algebra is called the binomial theorem. If instead of 1 + x we use x + a, we get

(x + a)ⁿ = xⁿ + 1ₙaxⁿ⁻¹ + 2ₙa²xⁿ⁻² + 3ₙa³xⁿ⁻³ + ... + nₙaⁿ

We can make the same table in another form. If we take a row of ciphers beginning with unity, and setting down the first, add the next, and then the next, and so on, and then repeat the process with one step less, and then again with one step less, we have the following:

1 0 0 0 0 0 0

1 1 1 1 1 1 1

1 2 3 4 5 6

1 3 6 10 15

1 4 10 20

1 5 15

1 6

1

In the oblique columns we see 1 1, 1 2 1, 1 3 3 1, &c. the same as in the original table, and formed by the same additions. If, before making the additions, we had always multiplied by a, we should have got the several components of the powers of 1 + a, thus,

1 0 0 0 0

1 a a² a³ a⁴

1 2a 3a² 4a³

1 3a 6a²

1 4a

1

where the oblique columns 1 + a, 1 + 2a + a², 1 + 3a + 3a² + a³, &c., give the several powers of 1 + a. If instead of beginning with 1, 0, 0, &c. we had begun with p, 0, 0, &c. we should have got p, p × 4a, p × 6a², &c. at the bottom of the several columns; and if we had written at the top x⁴, x³, x², x, 1, we should have had all the materials for forming p(x + a)⁴ by multiplying the terms at the top and bottom of each column together, and adding the results.

Suppose we follow this mode of forming p(x + a)³ + q(x + a)² + r(x + a) + s.

x³ x² x 1 x² x 1 x 1 1

p 0 0 0 q 0 0 r 0 1

p pa pa² pa³ q qa qa² r ra

p 2pa 3pa² q 2qa r

p 3pa q

p

px³ + 3pax² + 3pa²x + pa³ + qx² + 2qax + qa² + rx + ra + s

= px³ + (3pa + q)x² + (3pa² + 2qa + r)x + pa³ + qa² + ra + s

Now, observe that all this might be done in one process, by entering q, r, and s under their proper powers of x in the first process, as follows

x³ x² x 1

p q r s

p pa + q pa² + qa + r pa³ + qa² + ra + s

p 2pa + q 3pa² + 2qa + r

p 3pa + q

p

This process[65] is the one used in Appendix XI., with the slight alteration of varying the sign of the last letter, and making subtractions instead of additions in the last column. As it stands, it is the most convenient mode of writing x + a instead of x in a large class of algebraical expressions. For instance, what does 2x⁵ + x⁴ + 3x² + 7x + 9 become when x + 5 is written instead of x? The expression, made complete, is,

2x⁵ + 1x⁴ + 0x³ + 3x² + 7x + 9

1 0 3 7 9

2 11 55 278 1397 6994

2 21 160 1078 6787

2 31 315 2653

2 41 520

2 51

Answer, 2x⁵ + 51x⁴ + 520x³ + 2653x² + 6787x + 6994.

APPENDIX XI. ON HORNER’S METHOD OF SOLVING EQUATIONS.

The rule given in this chapter is inserted on account of its excellence as an exercise in computation. The examples chosen will require but little use of algebraical signs, that they may be understood by those who know no more of algebra than is contained in the present work.

To solve an equation such as

2x⁴ + x² - 3x = 416793,

or, as it is usually written,

2x⁴ + x² - 3x - 416793 = 0,

we must first ascertain by trial not only the first figure of the root, but also the denomination of it: if it be a 2, for instance, we must know whether it be 2, or 20, or 200, &c., or ·2, or ·02, or ·002, &c. This must be found by trial; and the shortest way of making the trial is as follows: Write the expression in its complete form. In the preceding case the form is not complete, and the complete form is

2x⁴ + 0x³ + 1x² - 3x - 416793.

To find what this is when x is any number, for instance, 3000, the best way is to take the first multiplier (2), multiply it by 3000, and take in the next multiplier (0), multiply the result by 3000, and take in the next multiplier (1), and so on to the end, as follows:

2 × 3000 + 0 = 6000; 6000 × 3000 + 1 = 18000001

18000001 × 3000 - 3 = 54000002997

54000002997 × 3000 - 416793 = 162000008574207

Now try the value of the above when x = 30. We have then, for the steps, 60 (2 × 30 + 0), 1801, 54027, and lastly,

1620810 - 416793,

or x = 30 makes the first terms greater than 416793. Now try x = 20 which gives 40, 801, 16017, and lastly,

320340 - 416793,

or x = 20 makes the first terms less than 416793. Between 20 and 30, then, must be a value of x which makes 2x⁴ + x²-3x equal to 416793. And this is the preliminary step of the process.

Having got thus far, write down the coefficients +2, 0, +1,-3, and -416793, each with its proper algebraical sign, except the last, in which let the sign be changed. This is the most convenient way when the last sign is-. But if the last sign be +, it may be more convenient to let it stand, and change all which come before. Thus, in solving x³-12x + 1 = 0, we might write

-1 0 +12 1

whereas in the instance before us, we write

+2 0 +1 -3 416793

Having done this, take the highest figure of the root, properly named, which is 2 tens, or 20. Begin with the first column, multiply by 20, and join it to the number in the next column; multiply that by 20, and join it to the number in the next column; and so on. But when you come to the last column, subtract the product which comes out of the preceding column, or join it to the last column after changing its sign. When this has been done, repeat the process with the numbers which now stand in the columns, omitting the last, that is, the subtracting step; then repeat it again, going only as far as the last column but two, and so on, until the columns present a set of rows of the following appearance:

a b c d e

f g h i

k l m

n o

p

to the formation of which the following is the key:

f = 20a + b,

g = 20f + c,

h = 20g + d,

i = e - 20h,

k = 20a + f,

l = 20k + g,

m = 20l + h,

n = 20a + k,

o = 20n + l,

p = 20a + n.

We call this Horner’s Process, from the name of its inventor. The result is as follows:

2 0 1 -3 416793 (20

40 801 16017 96453

80 2401 64037

120 4801

160

We have now before us the row

2 160 4801 64037 96453

which furnishes our means of guessing at the next, or units’ figure of the root.

Call the last column the dividend, the last but one the divisor, and all that come before antecedents. See how often the dividend contains the divisor; this gives the guess at the next figure. The guess is a true one,[66] if, on applying Horner’s process, the divisor result, augmented as it is by the antecedent processes, still go as many times in the dividend. For example, in the case before us, 96453 contains 64037 once; let 1 be put on its trial. Horner’s process is found to succeed, and we have for the second process,

2 160 4801 64037 96453

162 4963 69000 27453

164 5127 74127

166 5293

168

As soon as we come to the fractional portion of the root, the process assumes a more[67] methodical form.

The equation being of the fourth degree, annex four ciphers to the dividend, three to the divisor, two to the antecedent, and one to the previous antecedent, leaving the first column as it is; then find the new figure by the dividend and divisor, as before,[68] and apply Horner’s process. Annex ciphers to the results, as before, and proceed in the same way. The annexing of the ciphers prevents our having any thing to do with decimal points, and enables us to use the quotient-figures without paying any attention to their local values. The following exhibits the whole process from the beginning, carried as far as it is here intended to go before beginning the contraction, which will give more figures, as in the rule for the square root. The following, then, is the process as far as one decimal place:

2 0 1 -3 416793(213

40 801 16017 96453

80 2401 64037 274530000

120 4801 69000 47339778

160 4963 74127000

162 5127 75730074

164 529300 77348376

166 534358

1680 539434

1686 544528

1692

1698

1704

If we now begin the contraction, it is good to know beforehand on what number of additional root-figures we may reckon. We may be pretty certain of having nearly as many as there are figures in the divisor when we begin to contract—one less, or at least two less. Thus, there being now eight figures in the divisor, we may conclude that the contraction will give us at least six more figures. To begin the contraction, let the dividend stand, cut off one figure from the divisor, two from the column before that, three from the one before that, and so on. Thus, our contraction begins with

0002 1 704 5445 28 7734837 6 47339778

The first column is rendered quite useless here. Conduct the process as before, using only the figures which are not cut off. But it will be better to go as far as the first figure cut off, carrying from the second figure cut off. We shall then have as follows:

1 704 5445 28 7734837 6 47339778(6

5445 5 7767570 6 734354

5465 7 7800364 8

5475 9

At the next contraction the column 1|704 becomes |001704, and is quite useless. The next step, separately written (which is not, however, necessary in working), is

54 759 780036 48 734354(0

Here the dividend 734354 does not contain the divisor 780036, and we, therefore, write 0 as a root figure and make another contraction, or begin with

54759 78003 648 734354(9

78008 5 32277

78013 4

At the next contraction the first column becomes |0054759, and is quite useless, so that the remainder of the process is the contracted division.

7801 34) 32277 (4137

1072

292

58

3

and the root required is 21·36094137.

I now write down the complete process for another equation, one root of which lies between 3 and 4: it is

x³ - 10x + 1 = 0

1 0 -10 -1(3·111039052073099 0796

3 -1 2000

6 1700 209000

9 0 1791 19769000

9 1 188300 743369000000

9 2 189231 172311710273000

9 30 19016300 991247447681

9 31 19025631 39462875420

9 32 1903496300 0 0 1391491559

9 33 0 1903524299 0 9 58993123

9 33 1 1903552298 2 7 0 0 1886047

9 33 2 1903560698 0 5 9 1 172835

9 33 30 0 1903569097 8 5 6 3 1515

9 33 30 3 1903569144 5 2 2 183

9 33 30 6 1903569191 1 8 8 12

9 33 30 90 1903569193 0 6 1

9 33 30 99 1903569194 9 3

9 33 31 08

09 33 31 17

The student need not repeat the rows of figures so far as they come under one another: thus, it is not necessary to repeat 190356. But he must use his own discretion as to how much it would be safe for him to omit. I have set down the whole process here as a guide.

The following examples will serve for exercise:

1. 2x³ - 100x - 7 = 0

x = 7·10581133.

2. x⁴ + x³ + x² + x = 6000

x = 8·531437726.

3. x³ + 3x² - 4x - 10 = 0

x = 1·895694916504.

4. x³ + 100x² - 5x - 2173 = 0

x = 4·582246071058464.

_

5. ∛2 = 1·259921049894873164767210607278.[69]

6. x³ - 6x = 100

x = 5·071351748731.

7. x³ + 2x² + 3x = 300

x = 5·95525967122398.

8. x³ + x = 1000

x = 9·96666679.

9. 27000x³ + 27000x = 26999999

x = 9·9666666.....

10. x³ - 6x = 100

x = 5·0713517487.

11. x⁵ - 4x⁴ + 7x³ - 863 = 0

x = 4·5195507.

12. x³ - 20x + 8 = 0

x = 4·66003769300087278.

13. x³ + x² + x - 10 = 0

x = 1·737370233.

14. x³ - 46x² - 36x + 18 = 0

x = 46·7616301847,

or x = ·3471623192.

15. x³ + 46x² - 36x - 18 = 0

x = 1·1087925037.

16. 8991x³ - 162838x² + 746271x - 81000 = 0

x = ·111222333444555....

17. 729x³ - 486x² + 99x - 6 = 0

x = ·1111..., or ·2222..., or ·3333....

18. 2x³ + 3x² - 4x = 500

x = 5·93481796231515279.

19. x³ + 2x² + x - 150 = 0

x = 4·6684090145541983253742991201705899.

20. x³ + x = x² + 500

x = 8·240963558144858526963.

21. x³ + 2x² + 3x - 10000 = 0

x = 20·852905526009.

22. x⁵ - 4x - 2000 = 0

x = 4·581400362.

23. 10x³ - 33x² - 11x - 100 = 0

x = 4·146797808584278785.

24. x⁴ + x³ + x² + x = 127694

x = 18·64482373095.

25. 10x³ + 11x² + 12x = 100000

x = 21·1655995554508805.

26. x³ + x = 13

x = 2·209753301208849.

27. x³ + x² - 4x - 1600 = 0

x = 11·482837157.

28. x³ - 2x = 5

x = 2·094551481542326591482386540579302963857306105628239.

29. x⁴ - 80x³ + 24x² - 6x - 80379639 = 0

x = 123.[70]

30. x³ - 242x² - 6315x + 2577096 = 0

x = 123.[71]

31. 2x⁴ - 3x³ + 6x - 8 = 0

x = 1·414213562373095048803.[72]

32. x⁴ - 19x³ + 132x² - 302x + 200 = 0

x = 1·02804, or 4, or 6·57653, or 7·39543[73].

33. 7x⁴ - 11x³ + 6x² + 5x = 215

x = 2·70648049385791.[74]

34. 7x⁵ + 6x⁴ + 5x³ + 4x² + 3x = 11

x = ·770768819622658522379296505.[75]

35. 4x⁶ + 7x⁵ + 9x⁴ + 6x³ + 5x² + 3x = 792

x = 2·052042176879605365214043401281201973460275599545541724214.[76]

36. 2187x⁴ - 2430x³ + 945x² - 150x + 8 = 0

x = ·1111...., or ·2222...., or ·3333...., or ·4444....

APPENDIX XII. RULES FOR THE APPLICATION OF ARITHMETIC TO GEOMETRY.

The student should make himself familiar with the most common terms of geometry, after which the following rules will present no difficulty. In them all, it must be understood, that when we talk of multiplying one line by another, we mean the repetition of one line as often as there are units of a given kind, as feet or inches, in another. In any other sense, it is absurd to talk of multiplying a quantity by another quantity. All quantities of the same kind should be represented in numbers of the same unit; thus, all the lines should be either feet and decimals of a foot, or inches and decimals of an inch, &c. And in whatever unit a length is represented, a surface is expressed in the corresponding square units, and a solid in the corresponding cubic units. This being understood, the rules apply to all sorts of units.

To find the area of a rectangle. Multiply together the units in two sides which meet, or multiply together two sides which meet; the product is the number of square units in the area. Thus, if 6 feet and 5 feet be the sides, the area is 6 × 5, or 30 square feet. Similarly, the area of a square of 6 feet long is 6 × 6, or 36 square feet (234).

To find the area of a parallelogram. Multiply one side by the perpendicular distance between it and the opposite side; the product is the area required in square units.

To find the area of a trapezium.[77] Multiply either of the two sides which are not parallel by the perpendicular let fall upon it from the middle point of the other.

To find the area of a triangle. Multiply any side by the perpendicular let fall upon it from the opposite vertex, and take half the product. Or, halve the sum of the three sides, subtract the three sides severally from this half sum, multiply the four results together, and find the square root of the product. The result is the number of square units in the area; and twice this, divided by either side, is the perpendicular distance of that side from its opposite vertex.

To find the radius of the internal circle which touches the three sides of a triangle. Divide the area, found in the last paragraph, by half the sum of the sides.

Given the two sides of a right-angled triangle, to find the hypothenuse. Add the squares of the sides, and extract the square root of the sum.

Given the hypothenuse and one of the sides, to find the other side. Multiply the sum of the given lines by their difference, and extract the square root of the product.

To find the circumference of a circle from its radius, very nearly. Multiply twice the radius, or the diameter, by 3·1415927, taking as many decimal places as may be thought necessary. For a rough computation, multiply by 22 and divide by 7. For a very exact computation, in which decimals shall be avoided, multiply by 355 and divide by 113. See (131), last example.

To find the arc of a circular sector, very nearly, knowing the radius and the angle. Turn the angle into seconds,[78] multiply by the radius, and divide the product by 206265. The result will be the number of units in the arc.

To find the area of a circle from its radius, very nearly. Multiply the square of the radius by 3·1415927.

To find the area of a sector, very nearly, knowing the radius and the angle. Turn the angle into seconds, multiply by the square of the radius, and divide by 206265 × 2, or 412530.

To find the solid content of a rectangular parallelopiped. Multiply together three sides which meet: the result is the number of cubic units required. If the figure be not rectangular, multiply the area of one of its planes by the perpendicular distance between it and its opposite plane.

To find the solid content of a pyramid. Multiply the area of the base by the perpendicular let fall from the vertex upon the base, and divide by 3.

To find the solid content of a prism. Multiply the area of the base by the perpendicular distance between the opposite bases.

To find the surface of a sphere. Multiply 4 times the square of the radius by 3·1415927.

To find the solid content of a sphere. Multiply the cube of the radius by 3·1415927 × ⁴/₃, or 4·18879.

To find the surface of a right cone. Take half the product of the circumference of the base and slanting side. To find the solid content, take one-third of the product of the base and the altitude.

To find the surface of a right cylinder. Multiply the circumference of the base by the altitude. To find the solid content, multiply the area of the base by the altitude.

The weight of a body may be found, when its solid content is known, if the weight of one cubic inch or foot of the body be known. But it is usual to form tables, not of the weights of a cubic unit of different bodies, but of the proportion which these weights bear to some one amongst them. The one chosen is usually distilled water, and the proportion just mentioned is called the specific gravity. Thus, the specific gravity of gold is 19·362, or a cubic foot of gold is 19·362 times as heavy as a cubic foot of distilled water. Suppose now the weight of a sphere of gold is required, whose radius is 4 inches. The content of this sphere is 4 × 4 × 4 × 4·1888, or 268·0832 cubic inches; and since, by (217), each cubic inch of water weighs 252·458 grains, each cubic inch of gold weighs 252·458 × 19·362, or 4888·091 grains; so that 268·0832 cubic inches of gold weigh 268·0832 × 4888·091 grains, or 227½ pounds troy nearly. Tables of specific gravities may be found in most works of chemistry and practical mechanics.

The cubic foot of water is 908·8488 troy ounces, 75·7374 troy pounds, 997·1369691 averdupois ounces, and 62·3210606 averdupois pounds. For all rough purposes it will do to consider the cubic foot of water as being 1000 common ounces, which reduces tables of specific gravities to common terms in an obvious way. Thus, when we read of a substance which has the specific gravity 4·1172, we may take it that a cubic foot of the substance weighs 4117 ounces. For greater correctness, diminish this result by 3 parts out of a thousand.

THE END.

WALTON AND MABERLY’S

CATALOGUE OF EDUCATIONAL WORKS, AND WORKS IN SCIENCE AND GENERAL LITERATURE.

ENGLISH.

Dr. R. G. Latham. The English Language. Fourth Edition. 2 vols. 8vo. £1 8s. cloth.

Latham’s Elementary English Grammar, for the Use of Schools. Eighteenth thousand. Small 8vo. 4s. 6d. cloth.

Latham’s Hand-book of the English Language, for the Use of Students of the Universities and higher Classes of Schools. Third Edition. Small 8vo. 7s. 6d. cloth.

Latham’s Logic in its Application to Language. 12mo. 6s. cloth.

Latham’s First English Grammar, adapted for general use.By Dr. R. G. Latham and an Experienced Teacher. Fcap. 8vo.

Latham’s History and Etymology of the English Language, for the Use of Classical Schools. Second Edition, revised. Fcap. 8vo. 1s. 6d. cl.

Mason’s English Grammar, including the Principles of Grammatical Analysis. 12mo. 3s. 6d. cloth.

Mason’s Cowper’s Task Book I. (the Sofa), with Notes on the Analysis and Parsing. Crown 8vo. 1s. 6d. cloth.

Abbott’s First English Reader. Third Edition. 12mo., with Illustrations. 1s. cloth, limp.

Abbott’s Second English Reader. Third Edition. 12mo. 1s. 6d. cloth, limp.

GREEK.

Greenwood’s Greek Grammar, including Accidence, Irregular Verbs, and Principles of Derivation and Composition; adapted to the System of Crude Forms. Small 8vo. 5s. 6d. cloth.

Kühner’s New Greek Delectus; being Sentences for Translation from Greek into English, and English into Greek; arranged in a systematic Progression. Translated and Edited by the late Dr. Alexander Allen. Fourth Edition, revised. 12mo. 4s. cloth.

Gillespie’s Greek Testament Roots, in a Selection of Texts, giving the power of Reading the whole Greek Testament without difficulty. With Grammatical Notes, and a Parsing Lexicon associating the Greek Primitives with English Derivatives. Post 8vo. 7s. 6d. cloth.

Robson’s Constructive Exercises for Teaching the Elements of the Greek Language, on a system of Analysis and Synthesis, with Greek Reading Lessons and copious Vocabularies. 12mo., pp. 408. 7s. 6d. cloth.

Robson’s First Greek Book. Exercises and Reading Lessons with Copious Vocabularies. Being the First Part of the “Constructive Greek Exercises.” 12mo. 3s. 6d. cloth.

The London Greek Grammar. Designed to exhibit, in small Compass, the Elements of the Greek Language. Sixth Edition. 12mo. 1s. 6d. cloth.

Hardy and Adams’s Anabasis of Xenophon. Expressly for Schools. With Notes, Index of Names, and a Map. 12mo. 4s. 6d. cloth.

Smith’s Plato. The Apology of Socrates, The Crito, and part of the Phaedo; with Notes in English from Stallbaum, Schleiermacher’s Introductions, and his Essay on the Worth of Socrates as a Philosopher. Edited by Dr. Wm. Smith, Editor of the Dictionary of Greek and Roman Antiquities, &c. Third Edition. 12mo. 5s. cloth.

LATIN.

New Latin Reading Book; consisting of Short Sentences, Easy Narrations, and Descriptions, selected from Caesar’s Gallic War; in Systematic Progression. With a Dictionary. Second Edition, revised. 12mo. 2s. 6d.

Allen’s New Latin Delectus; being Sentences for Translation from Latin into English, and English into Latin; arranged in a systematic Progression. Fourth Edition, revised. 12mo. 4s. cloth.

The London Latin Grammar;; including the Eton Syntax and Prosody in English, accompanied with Notes. Sixteenth Edition. 12mo. 1s. 6d.

Robson’s Constructive Latin Exercises, for teaching the Elements of the Language on a System of Analysis and Synthesis; with Latin Reading Lessons and Copious Vocabularies. Third and Cheaper Edition, thoroughly revised. 12mo. 4s. 6d. cloth.

Robson’s First Latin Reading Lessons. With Complete Vocabularies. Intended as an Introduction to Caesar. 12mo. 2s. 6d. cloth.

Smith’s Tacitus; Germania, Agricola, and First Book of the Annals. With English Notes, original and selected, and Bötticher’s remarks on the style of Tacitus. Edited by Dr. Wm. Smith, Editor of the Dictionary of Greek and Roman Antiquities, etc. Third Edition, greatly improved. 12mo. 5s.

Caesar. Civil War. Book I. With English Notes for the Use of Students preparing for the Cambridge School Examination. 12mo. 1s. 6d.

Terence. Andria. With English Notes, Summaries, and Life of Terence. By Newenham Travers, B.A., Assistant-Master in University College School. Fcap.8vo. 3s. 6d.

HEBREW.

Hurwitz’s Grammar of the Hebrew Language. Fourth Edition. 8vo. 13s. cloth. Or in Two Parts, sold separately:—Elements. 4s. 6d. cloth. Etymology and Syntax. 9s. cloth.

FRENCH.

Merlet’s French Grammar. By P. F. Merlet, Professor of French in University College, London. New Edition. 12mo. 5s. 6d. bound. Or sold in Two Parts:—Pronunciation and Accidence, 3s. 6d.; Syntax, 3s. 6d. (Key, 3s. 6d.)

Merlet’s Le Traducteur; Selections, Historical, Dramatic, and Miscellaneous, from the best French Writers, on a plan calculated to render reading and translation peculiarly serviceable in acquiring the French Language; accompanied by Explanatory Notes, a Selection of Idioms, etc. Fourteenth Edition. 12mo. 5s. 6d. bound.

Merlet’s Exercises on French Composition. Consisting of Extracts from English Authors to be turned into French; with Notes indicating the Differences in Style between the two Languages. A List of Idioms, with Explanations, Mercantile Terms and Correspondence, Essays, etc. 12mo. 3s. 6d.

Merlet’s French Synonymes, explained in Alphabetical Order. Copious Examples (from the “Dictionary of Difficulties”). 12mo. 2s. 6d.

Merlet’s Aperçu de la Litterature Française. 12mo. 2s. 6d.

Merlet’s Stories from French Writers; in French and English Interlinear (from Merlet’s “Traducteur”). Second Edition. 12mo. 2s. cl.

Lombard De Luc’s Classiques Français, à l’Usage de la Jeunesse Protestante; or, Selections from the best French Classical Works, preceded by Sketches of the Lives and Times of the Writers. 12mo. 3s. 6d. cloth.

ITALIAN.

Smith’s First Italian Course; being a Practical and Easy Method of Learning the Elements of the Italian Language. Edited from the German of Filippi, after the method of Dr. Ahn. 12mo. 3s. 6d. cloth.

INTERLINEAR TRANSLATIONS.

Locke’s System of Classical Instruction.

Interlinear Translations 1s. 6d. each.

Latin.

Phaedrus’s Fables of Æsop.

Virgil’s Æneid. Book I.

Parsing Lessons to Virgil.

Caesar’s Invasion of Britain.

Greek.

Lucian’s Dialogues. Selections.

Homer’s Iliad. Book I.

Xenophon’s Memorabilia. Book I.

Herodotus’s Histories. Selections.

French.

Sismondi; the Battles of Cressy and Poictiers.

German.

Stories from German Writers.

Also, to accompany the Latin and Greek Series.

The London Latin Grammar. 12mo. 1s. 6d.

The London Greek Grammar. 12mo. 1s. 6d.

HISTORY, MYTHOLOGY, AND ANTIQUITIES.

Creasy’s (Professor) History of England. With Illustrations. One Volume. Small 8vo. Uniform with Schmitz’s “History of Rome,” and Smith’s “History of Greece.” (Preparing).

Schmitz’s History of Rome, from the Earliest Times to the Death of Commodus, a.d. 192. Ninth Edition. One Hundred Engravings. 12mo. 7s. 6d. cloth.

Smith’s History of Greece, from the Earliest Times to the Roman Conquest. With Supplementary Chapters on the History of Literature and Art. New Edition. One Hundred Engravings on Wood. Large 12mo. 7s. 6d. cloth.

Smith’s Smaller History of Greece. With Illustrations. Fcp. 8vo. 3s. 6d. cloth.

Smith’s Dictionary of Greek and Roman Antiquities. By various Writers. Second Edition. Illustrated by Several Hundred Engravings on Wood. One thick volume, medium 8vo. £2 2s. cloth.

Smith’s Smaller Dictionary of Greek and Roman Antiquities. Abridged from the larger Dictionary. New Edition. Crown 8vo., 7s. 6d. cloth.

Smith’s Dictionary of Greek and Roman Biography and Mythology. By various Writers. Medium 8vo. Illustrated by numerous Engravings on Wood. Complete in Three Volumes. 8vo. £5 15s. 6d. cloth.

Smith’s New Classical Dictionary of Biography, Mythology, and Geography. Partly based on the “Dictionary of Greek and Roman Biography and Mythology.” Third Edition 750 Illustrations. 8vo. 18s. cloth.

Smith’s Smaller Classical Dictionary of Biography, Mythology, and Geography. Abridged from the larger Dictionary. Illustrated by 200 Engravings on Wood. New Edition. Crown 8vo. 7s. 6d. cloth.

Smith’s Dictionary of Greek and Roman Geography. By various Writers. Illustrated with Woodcuts of Coins, Plans of Cities, etc. Two Volumes 8vo. £4. cloth.

Niebuhr’s History of Rome. From the Earliest Times to the First Punic War. Fourth Edition. Translated by Bishop Thirlwall, Archdeacon Hare, Dr. Smith, and Dr. Schmitz. Three Vols. 8vo. £1 16s.

Niebuhr’s Lectures on the History of Rome. From the Earliest Times to the First Punic War. Edited by Dr. Schmitz. Third Edition. 8vo. 8s.

Newman (F. W.) The Odes of Horace. Translated into Unrhymed Metres, with Introduction and Notes. Crown 8vo. 5s. cloth.

Newman (F. W.) The Iliad of Homer, Faithfully translated into Unrhymed Metre. 1 vol. crown 8vo 6s. 6d. cloth.

Akerman’s Numismatic Manual, or Guide to the Collection and Study of Greek, Roman, and English Coins. Many Engravings. 8vo. £1 1s.

Ramsay’s (George) Principles of Psychology. 8vo. 1Os. 6d.

PURE MATHEMATICS.

De Morgan’s Elements of Arithmetic. Fifteenth Thousand. Royal 12mo. 5s. cloth.

De Morgan’s Trigonometry and Double Algebra. Royal 12mo. 7s. 6d. cloth.

Ellenberger’s Course of Arithmetic, as taught in the Pestalozzian School, Worksop. Post 8vo. 5s. cloth. ⁂ The Answers to the Questions in this Volume are now ready, price 1s. 6d.

Mason’s First Book of Euclid. Explained to Beginners. Fcap. 8vo. 1s. 9d.

Reiner’s Lessons on Form; or, An Introduction to Geometry, as given in a Pestalozzian School, Cheam, Surrey. 12mo. 3s. 6d.

Reiner’s Lessons on Number, as given in a Pestalozzian School, Cheam, Surrey. Master’s Manual, 5s. Scholar’s Praxis, 2s.

Tables of Logarithms Common and Trigonometrical to Five Places. Under the Superintendence of the Society for the Diffusion of Useful Knowledge. Fcap. 8vo. 1s. 6d.

Four Figure Logarithms and Anti-Logarithms. On a Card. Price 1s.

Barlow’s Tables of Squares, Cubes, Square Roots, Cube Roots, and Reciprocals of all Integer Numbers, up to 10,000. Royal 12mo. 8s.

MIXED MATHEMATICS.

Potter’s Treatise on Mechanics, for Junior University Students. By Richard Potter, M.A., Professor of Natural Philosophy in University College, London. Third Edition. 8vo. 8s. 6d.

Potter’s Treatise on Optics. Part I. All the requisite Propositions carried to First Approximations, with the construction of Optical Instruments, for Junior University Students. Second Edition. 8vo. 9s. 6d.

Potter’s Treatise on Optics. Part II. The Higher Propositions, with their application to the more perfect forms of Instruments. 8vo. 12s. 6d.

Potter’s Physical Optics; or, the Nature and Properties of Light. A Descriptive and Experimental Treatise. 100 Illustrations. 8vo. 6s. 6d.

Newth’s Mathematical Examples. A graduated series of Elementary Examples, in Arithmetic, Algebra, Logarithms, Trigonometry, and Mechanics. Crown 8vo. With Answers. 8s. 6d. cloth.

Sold also in separate Parts, without Answers:—

Arithmetic, 2s. 6d.

Algebra, 2s. 6d.

Trigonometry and Logarithms, 2s. 6d.

Mechanics, 2s. 6d.

Newth’s Elements of Mechanics, including Hydrostatics, with numerous Examples. By Samuel Newth, M.A., Fellow of University College, London. Second Edition. Large 12mo. 7s. 6d. cloth.

Newth’s First Book of Natural Philosophy; or an Introduction to the Study of Statics, Dynamics, Hydrostatics, and Optics, with numerous Examples. 12mo. 3s. 6d. cloth.

NATURAL PHILOSOPHY, ASTRONOMY, Etc.

Lardner’s Museum of Science and Art. Complete in 12 Single Volumes, 18s., ornamental boards; or 6 Double Ones, £1 1s., cl. lettered.

⁂ Also, handsomely half-bound morocco, 6 volumes, £1 11s. 6d.

Contents:—The Planets; are they inhabited Worlds?

Weather Prognostics. Popular Fallacies in Questions

of Physical Science. Latitudes and Longitudes. Lunar

Influences. Meteoric Stones and Shooting Stars. Railway

Accidents. Light. Common Things.—Air. Locomotion

in the United States. Cometary Influences. Common

Things.—Water. The Potter’s Art. Common Things.—Fire.

Locomotion and Transport, their Influence and Progress.

The Moon. Common Things.—The Earth. The Electric

Telegraph. Terrestrial Heat. The Sun. Earthquakes and

Volcanoes. Barometer, Safety Lamp, and Whitworth’s

Micrometric Apparatus. Steam. The Steam Engine. The

Eye. The Atmosphere. Time. Common Things.—Pumps.

Common Things.—Spectacles—The Kaleidoscope. Clocks

and Watches. Microscopic Drawing and Engraving. The

Locomotive. Thermometer. New Planets.—Leverrier and

Adams’s Planet. Magnitude and Minuteness. Common

Things.—The Almanack. Optical Images. How to Observe

the Heavens. Common Things.—The Looking Glass. Stellar

Universe. The Tides. Colour. Common Things.—Man.

Magnifying Glasses. Instinct and Intelligence. The Solar

Microscope. The Camera Lucida. The Magic Lantern. The

Camera Obscura. The Microscope. The White Ants; their

Manners and Habits. The Surface of the Earth, or First

Notions of Geography. Science and Poetry. The Bee. Steam

Navigation. Electro-Motive Power. Thunder, Lightning,

and the Aurora Borealis. The Printing Press. The Crust

of the Earth. Comets. The Stereoscope. The Pre-Adamite

Earth. Eclipses. Sound.

Lardner’s Animal Physics, or the Body and its Functions familiarly Explained. 520 Illustrations. 1 vol., small 8vo. 12s. 6d. cloth.

Lardner’s Animal Physiology for Schools (chiefly taken from the “Animal Physics”). 190 Illustrations. 12mo. 3s. 6d. cloth.

Lardner’s Hand-Book of Mechanics. 357 Illustrations. 1 vol., small 8vo., 5s.

Lardner’s Hand-Book of Hydrostatics, Pneumatics, and Heat. 292 Illustrations. 1 vol., small 8vo., 5s.

Lardner’s Hand-Book of Optics. 290 Illustrations. 1 vol., small 8vo., 5s.

Lardner’s Hand-Book of Electricity, Magnetism, and Acoustics. 395 Illustrations. 1 vol., small 8vo. 5s.

Lardner’s Hand-Book of Astronomy and Meteorology, forming a companion work to the “Hand-Book of Natural Philosophy.” 37 Plates, and upwards of 200 Illustrations on Wood. 2 vols., each 5s., cloth lettered.

Lardner’s Natural Philosophy for Schools. 328 Illustrations. 1 vol., large 12mo., 3s. 6d. cloth.

Lardner’s Chemistry for Schools. 170 Illustrations. 1 vol., large 12mo. 3s. 6d. cloth.

Pictorial Illustrations of Science and Art.

Large Printed Sheets,

each containing from 50 to 100 Engraved Figures.

Part I. 1s. 6d.

1. Mechanic Powers.

2. Machinery.

3. Watch and Clock Work.

Part II. 1s. 6d.

4. Elements of Machinery.

5. Motion and Force.

6. Steam Engine.

Part III. 1s. 6d.

7. Hydrostatics.

8. Hydraulics.

9. Pneumatics.

Lardner’s Popular Geology. (From “The Museum of Science and Art.”) 201 Illustrations. 2s. 6d.

Lardner’s Common Things Explained. Containing:

Air—Earth—Fire—Water—Time—The Almanack—Clocks and Watches—Spectacles—Colour—Kaleidoscope— Pumps—Man—The Eye—The Printing Press—The Potter’s Art—Locomotion and Transport—The Surface of the Earth, or First Notions of Geography. (From “The Museum of Science and Art.”) With 233 Illustrations. Complete, 5s., cloth lettered.

⁂ Sold also in Two Series, 2s. 6d. each.

Lardner’s Popular Physics. Containing:

Magnitude and Minuteness—Atmosphere—Thunder and Lightning—Terrestrial Heat—Meteoric Stones—Popular Fallacies— Weather Prognostics—Thermometer—Barometer—Safety Lamp—Whitworth’s Micrometric Apparatus—Electro-Motive Power—Sound—Magic Lantern—Camera Obscura—Camera Lucida—Looking Glass—Stereoscope—Science and Poetry. (From “The Museum of Science and Art.”) With 85 Illustrations. 2s. 6d. cloth lettered.

Lardner’s Popular Astronomy. Containing:

How to Observe the Heavens—Latitudes and Longitudes —TheEarth—The Sun—The Moon—The Planets: are they Inhabited?—The New Planets—Leverrier and Adams’s Planet—The Tides—Lunar Influences—and the Stellar Universe—Light—Comets—Cometary Influences—Eclipses—Terrestrial Rotation—Lunar Rotation—Astronomical Instruments. (From “The Museum of Science and Art.”) 182 Illustrations. Complete, 4s. 6d. cloth lettered.

⁂ Sold also in Two Series, 2s. 6d. and 2s.each.

Lardner on the Microscope. (From “The Museum of Science and Art.”) 1 vol. 147 Engravings. 2s.

Lardner on the Bee and White Ants; their Manners and Habits; with Illustrations of Animal Instinct and Intelligence. (From “The Museum of Science and Art.”) 1 vol. 135 Illustrations. 2s., cloth lettered.

Lardner on Steam and its Uses; including the Steam Engine and Locomotive, and Steam Navigation. (From “The Museum of Science and Art.”) 1 vol., with 89 Illustrations. 2s.

Lardner on the Electric Telegraph, Popularised. With 100 Illustrations. (From “The Museum of Science and Art.”) 12mo., 250 pages. 2s., cloth lettered.

⁂ The following Works from “Lardner’s Museum of Science and Art,” may also be had arranged as described, handsomely half bound morocco, cloth sides.

Common Things. Two series in one vol. 7s. 6d.

Popular Astronomy. Two series in one vol. 7s. 0d.

Electric Telegraph, with Steam and its Uses. In one vol. 7s. 0d.

Microscope and Popular Physics. In one vol. 7s. 0d.

Popular Geology, and Bee and White Ants. In one vol. 7s. 6d.

Lardner on the Steam Engine, Steam Navigation, Roads, and Railways. Explained and Illustrated. Eighth Edition. With numerous Illustrations. 1 vol. large 12mo. 8s. 6d.

A Guide to the Stars for every Night in the Year. In Eight Planispheres. With an Introduction. 8vo. 5s., cloth.

Minasi’s Mechanical Diagrams. For the Use of Lecturers and Schools. 15 Sheets of Diagrams, coloured, 15s., illustrating the following subjects: 1 and 2. Composition of Forces.—3. Equilibrium.—4 and 5. Levers.—6. Steelyard, Brady Balance, and Danish Balance.—7. Wheel and Axle.—8. Inclined Plane.—9, 10, 11. Pulleys.—12. Hunter’s Screw.—13 and 14. Toothed Wheels.—15. Combination of the Mechanical Powers.

LOGIC.

De Morgan’s Formal Logic; or, The Calculus of Inference, Necessary and Probable. 8vo. 6s. 6d.

Neil’s Art of Reasoning: a Popular Exposition of the Principles of Logic, Inductive and Deductive; with an Introductory Outline of the History of Logic, and an Appendix on recent Logical Developments, with Notes. Crown 8vo. 4s. 6d., cloth.

ENGLISH COMPOSITION.

Neil’s Elements of Rhetoric; a Manual of the Laws of Taste, including the Theory and Practice of Composition. Crown 8vo. 4s. 6d., cl.

DRAWING.

Lineal Drawing Copies for the earliest Instruction.Comprising upwards of 200 subjects on 24 sheets, mounted on 12 pieces of thick pasteboard, in a Portfolio. By the Author of “Drawing for Young Children.” 5s. 6d.

Easy Drawing Copies for Elementary Instruction. Simple Outlines without Perspective. 67 subjects, in a Portfolio. By the Author of “Drawing for Young Children.” 6s. 6d.

Sold also in Two Sets.

Set I. Twenty-six Subjects mounted on thick pasteboard, in a Portfolio. 3s. 6d.

Set II. Forty-one Subjects mounted on thick pasteboard, in a Portfolio. 3s. 6d.

The copies are sufficiently large and bold to be drawn from by forty or fifty children at the same time.

SINGING.

A Musical Gift from an Old Friend, containing Twenty-four New Songs for the Young. By W. E Hickson, author of the Moral Songs of “The Singing Master.” 8vo. 2s. 6d.

The Singing Master. Containing First Lessons in Singing, and the Notation of Music; Rudiments of the Science of Harmony; The First Class Tune Book; The Second Class Tune Book; and the Hymn Tune Book. Sixth Edition. 8vo. 6s., cloth lettered. Sold also in Five Parts, any of which may be had separately.

I.—First Lessons in Singing and the Notation of Music. Containing Nineteen Lessons in the Notation and Art of Reading Music, as adapted for the Instruction of Children, and especially for Class Teaching, with Sixteen Vocal Exercises, arranged as simple two-part harmonies. 8vo. 1s., sewed.

II.—Rudiments of the Science of Harmony or Thorough Bass. Containing a general view of the principles of Musical Composition, the Nature of Chords and Discords, mode of applying them, and an Explanation of Musical Terms connected with this branch of Science. 8vo. 1s., sewed.

III.—The First Class Tune Book. A Selection of Thirty Single and Pleasing Airs, arranged with suitable words for young children. 8vo. 1s., sewed.

IV.—The Second Class Tune Book. A Selection of Vocal Music adapted for youth of different ages, and arranged (with suitable words) as two or three-part harmonies. 8vo, 1s. 6d.

V.—The Hymn Tune Book. A Selection of Seventy popular Hymn and Psalm Tunes, arranged with a view of facilitating the progress of Children learning to sing in parts. 8vo. 1s. 6d.

⁂ The Vocal Exercises, Moral Songs, and Hymns, with the Music, may also be had, printed on Cards, price Twopence each Card, or Twenty-five for Three Shillings.

CHEMISTRY.

Gregory’s Hand-Book of Chemistry. For the use of Students. By William Gregory, M.D., late Professor of Chemistry in the University of Edinburgh. Fourth Edition, revised and enlarged. Illustrated by Engravings on Wood. Complete in One Volume. Large 12mo. 18s. cloth.

⁂ The Work may also be had in two Volumes, as under.

Inorganic Chemistry. Fourth Edition, revised and enlarged. 6s. 6d. cloth.

Organic Chemistry. Fourth Edition, very carefully revised, and greatly enlarged. 12s., cloth.(Sold separately.)

Chemistry for Schools. By Dr. Lardner.190 Illustrations. Large 12mo. 3s. 6d. cloth.

Liebig’s Familiar Letters on Chemistry, in its Relations to Physiology, Dietetics, Agriculture, Commerce, and Political Economy. Fourth Edition, revised and enlarged, with additional Letters. Edited by Dr. Blyth. Small 8vo. 7s. 6d. cloth.

Liebig’s Letters on Modern Agriculture. Small 8vo. 6s.

Liebig’s Principles of Agricultural Chemistry; with Special Reference to the late Researches made in England. Small 8vo. 3s. 6d., cloth.

Liebig’s Chemistry in its Applications to Agriculture and Physiology. Fourth Edition, revised. 8vo. 6s. 6d., cloth.

Liebig’s Animal Chemistry; or, Chemistry in its Application to Physiology and Pathology. Third Edition. Part I. (the first half of the work). 8vo. 6s. 6d., cloth.

Liebig’s Hand-Book of Organic Analysis; containing a detailed Account of the various Methods used in determining the Elementary Composition of Organic Substances. Illustrated by 85 Woodcuts. 12mo. 5s., cloth.

Bunsen’s Gasometry; comprising the Leading Physical and Chemical Properties of Gases, together with the Methods of Gas Analysis. Fifty-nine Illustrations. 8vo. 8s. 6d., cloth.

Wöhler’s Hand-Book of Inorganic Analysis; One Hundred and Twenty-two Examples, illustrating the most important processes for determining the Elementary composition of Mineral substances. Edited by Dr. A. W. Hofmann, Professor in the Royal College of Chemistry. Large 12mo.

Parnell on Dyeing and Calico Printing. (Reprinted from Parnell’s “Applied Chemistry in Manufactures, Arts, and Domestic Economy, 1844.”) With Illustrations. 8vo. 7s., cloth.

GENERAL LITERATURE.

De Morgans Book of Almanacs. With an Index of Reference by which the Almanac may be found for every Year, whether in Old Style or New, from any Epoch, Ancient or Modern, up to a.d. 2000. With means of finding the Day of New or Full Moon, from b.c. 2000 to a.d. 2000. 5s., cloth lettered.

Guesses at Truth. By Two Brothers. New Edition. With an Index. Complete 1 vol. Small 8vo. Handsomely bound in cloth with red edges. 10s. 6d.

Rudall’s Memoir of the Rev. James Crabb; late of Southampton. With Portrait. Large 12mo., 6s., cloth.

Herschell (R. H.) The Jews; a brief Sketch of their Present State and Future Expectations. Fcap. 8vo. 1s. 6d., cloth.

Footnotes: