throughout all the columns, and set down the whole sum of the last or left-hand column. Thus:—
8976
4368
⎺⎺⎺⎺⎺
13344
Adding the units, 8 and 6 are 14. Therefore write down 4 and add 1 to the tens column.
Adding the tens, 1 and 6 and 7 are 14. Therefore write down 4 and add 1 to the hundreds column.
Adding the hundreds, 1 and 3 and 9 are 13. Therefore write down 3 and add 1 to the thousands column.
Adding the thousands, 1 and 4 and 8 are 13.
N.B.—The same rule evidently applies if there are more than two lines of figures to be added together.
6. Test of Correctness.—There are various methods by which the correctness of the process of addition may be tested.
Perhaps the most convenient test is to add the numbers together in the reverse order; that is, to commence with the top line instead of the bottom. If the second result be the same as the first, the work may be presumed to be right; for it is highly improbable that the same error will have been made in performing the operation in two different orders.
Exercise 4.
1. Add together the following sets of numbers:—
1. 75234 + 41015 + 19075 + 176.
2. 85064 + 9035 + 72358 + 919.
3. 1500267 + 45085 + 4652 + 4780400 + 90276 + 89760841.
4. 40702185 + 67070420 + 670856 + 4230823 + 750642 + 8790845.
5. 756 + 849 + 934 + 680 + 720 + 843 + 657680 + 989876498 + 8045685 + 807266780.
6. 432678902 + 310046734 + 2167005 + 327861 + 293000428.
7. 493742 + 56710607 + 23461 + 400072 + 6811004 + 8999003 + 26501.
8. 16075 + 250763 + 7561 + 830654 + 293106 + 2537104 + 31 725.
9. 142857 + 428571 + 285714 + 857142 + 571428 + 714285 + 142857.
10. 9034781 + 57 + 4897 + 309 + 587896 + 369875625 + 1876 + 398 + 79 + 8.
2. Add together the following numbers:—
Twenty-three thousand three hundred and forty-nine; seven thousand two hundred and seven; three hundred and twenty-five; five millions two hundred and fifty-three; fifty-six billions three hundred and nine millions five hundred and thirty-one thousand six hundred and nine; four thousand and seventeen millions; four thousand and four.
3. Find the sum of all the numbers from 1 to 100.
4. Arrange the nine digits in the form of a square, that is, in three rows of three figures each, so that when the columns are added vertically (up and down), horizontally (from side to side), or diagonally (from corner to corner), they will still produce the same sum.
5. In the following square, taken from Professor De Morgan's "Elements of Arithmetic," the columns added vertically, horizontally, or diagonally, will all produce the same sum, thus affording twenty-four different exercises in addition:—
| 2016 | 4212 | 1656 | 3852 | 1296 | 3492 | 936 | 3132 | 576 | 2772 | 216 |
| 252 | 2052 | 4248 | 1692 | 3888 | 1332 | 3528 | 972 | 3168 | 612 | 2412 |
| 2448 | 288 | 2088 | 4284 | 1728 | 3924 | 1368 | 3564 | 1008 | 2808 | 648 |
| 684 | 2484 | 324 | 2124 | 4320 | 1764 | 3960 | 1404 | 3204 | 1044 | 2844 |
| 2880 | 720 | 2520 | 360 | 2160 | 4356 | 1800 | 3600 | 1440 | 3240 | 1080 |
| 1116 | 2916 | 756 | 2556 | 396 | 2196 | 3996 | 1836 | 3636 | 1476 | 3276 |
| 3312 | 1152 | 2952 | 792 | 2592 | 36 | 2232 | 4032 | 1872 | 3672 | 1512 |
| 1548 | 3348 | 1188 | 2988 | 432 | 2628 | 72 | 2268 | 4068 | 1908 | 3708 |
| 3744 | 1584 | 3384 | 828 | 3024 | 468 | 2664 | 108 | 2304 | 4104 | 1944 |
| 1980 | 3780 | 1244 | 3420 | 864 | 3060 | 504 | 2700 | 144 | 2340 | 4140 |
| 4176 | 1620 | 3816 | 1260 | 3456 | 900 | 8096 | 540 | 2736 | 180 | 2376 |
6. The following is another example of the same kind, which will afford sixteen exercises on larger numbers than those in the preceding square:—
| 2177956 | 4652906 | 1583968 | 4058918 | 989980 | 3464930 | 395992 |
| 494990 | 2276954 | 4751904 | 1682966 | 4157916 | 1088978 | 2870942 |
| 2969940 | 593988 | 2375952 | 4850902 | 1781964 | 3563928 | 1187976 |
| 1286974 | 3068938 | 692986 | 2474950 | 4256914 | 1880962 | 3652926 |
| 3761924 | 1385972 | 3167936 | 98998 | 2573948 | 4355912 | 1979960 |
| 2078958 | 3860922 | 791984 | 3266934 | 197996 | 2672948 | 4454910 |
| 4553908 | 1484970 | 3959920 | 890982 | 3365932 | 296994 | 2771944 |
LESSONS IN BOTANY.—I.
INTRODUCTION.
At the outset we may as well state that by the term Botany we mean the science which teaches all about plants; such as their form, their aspect, the number and structure of their flowers, their seeds, and, in short, all that in any way relates to them. The word botany is derived from the Greek, in which language βοτάνη (bot'-a-ne), signifies a plant. Our friends the Germans call the study pflanzenlehre, plant-teaching; and, in our opinion, they are quite right to find a name for this and many other sciences out of their own language. We English might with great propriety do the same on many occasions, but it is not the custom.
Botany is a very interesting, no less than a very useful study, and it possesses over many others the advantage of being attended with no expense.
Inasmuch as botany is the science which teaches all about plants, the learner will agree that it is necessary to set out with precise notions as to what a plant is. Nothing would appear to be more easy than this; and easy enough it is when we take extreme cases: thus, for instance, no one would ever take an oak-tree for an animal, or a horse or an elephant for a vegetable; but there are certain beings whose characteristics are so little marked, that philosophers are to this day not agreed as to the division of nature to which they ought to be referred; in other cases, again, beings have been taken out of one classification and inserted under another; this remark applies to the sponge, which, although it grows attached to rocks under the sea, is now universally considered to be an animal, or, more properly speaking, the skeleton of an animal, the soft portions of which have been dissolved away.
The great Swedish naturalist Linné, better known by the Latin form of his name—Linnæus, adopted the following pithy designation of minerals, vegetables, and animals.
"Minerals," he said, "grow; plants grow and live; but animals grow, live, and feel." A very neatly turned expression this is, we must all allow, and the task would not be easy in few words to show wherein it is insufficient. Naturalists of the present day, however, do not consider it quite correct, and, what is more, naturalists own that their ingenuity has been unable to find a distinction which is quite correct: however, the following is perhaps more nearly correct than any other. Animals are those living beings which derive their nutriment from an internal cavity (the stomach), and vegetables are those living beings which absorb their nutriment from without.
SECTION I.—ON THE PRINCIPLES WHICH SERVE FOR THE CLASSIFICATION OF PLANTS.
Whatever may be the subject of our study it requires to be classified, classification being the very key-stone of order, without which our ideas become obscure and confused: therefore it is that even the least botanical amongst us, when speaking of vegetables, make a rough sort of classification for ourselves, usually dividing them into herbs, plants, bushes, or shrubs and trees. And for many common purposes this rough and ready distinction is sufficient; but it is not very correct, and therefore will not answer the purposes of a botanist.
To prove that the distinction is not correct, we will mention two cases in point, and we are sure the learner will accede to the justice of the remark. What would the reader term a myrtle
