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The lime is the European linden, so well known to us as a shade tree. In late June it is murmurous when in blossom, and it flowers profusely, scenting the whole air. Tennyson seems especially fond of this tree. The allusions to it are incessant, and always pertinent, as in the line,
“A myriad emeralds break from the ruby-budded lime.” Here is a line worth pages of ordinary description; it is from In Memoriam,
“When rosy plumelets tuft the larch." The fertile catkins, developed early, are crimson in flower, and appear midst the soft, new, feathery foliage. Unlike most Conifere, the larch has deciduous leaves.
Here is a picture of checkered shade :
“ Witch elms that counter-change the floor
Of this flat lawn with dusk and bright.”
The beginning of Part CI of the same poem gives us this wonderful painting, which makes less gifted observers envious :
“Unloved, that beech shall gather brown,
This maple burn itself away;
Unloved, the sunflower, shining fair,
May round with flowers her disk of seed,
And many a rose carnation feed
With summer spice the humming air." We now come to Maud, perhaps the most familiar as it is the most passionate of Tennyson's poems. It is one redolent with the scent of flowers. The very first lines embody a botanical description :
“ I hate the dreadful hollow behind the little wood,
Its lips in the field above are dabbled with blood-red heath.” This description will do for our American lilies :
" Woodland lilies,
Myriads blow together." Have we not seen them so in the meadows along our New England railroads in July and August? The couplet is almost as fine as Wordsworth's “ Host of golden daffodils.” The
“crimson-tipped daisy" of Burns is here made by the lover to do immortal homage :
“ Her feet have touched the meadows
And left the daisies rosy." Here is a different kind of figure, showing acute observation :
“ The dry-tongued laurel's pattering talk." Mind, this does not refer to our mountain laurel or Kalmia, but to some true laurel or bay. Listen to this plaint for the exiled trees of Syria, who “cannot sing the Lord's song in a strange land":
“Oh, art thou sighing for Lebanon,
Dark cedar, tho'thy limbs have been increased ?” In the ecstatic song of his triumphant wooing, the lover flings about his flowers with unstinting hand. The woodbine, the jessamine, the rose, the lily, the acacia, the daffodil, the violet, burst into joyous flower. Then comes a matchless stanza :
“The slender acacia would not shake
One long milk bloom on the tree;
As the pimpernel dozed on the lea.” The pimpernel, or poor man's weather glass, is the Anagallis arvensis, a tiny plant in sandy places on our coasts. The red, or rarely blue, flowers close when the sun is overcast. The verse above, then, foreshadows gloom.
The passion flower and the larkspur also appear in this melodious song. Even in that magnificent Ode to the Duke of Wellington we have the use of flowers :
“ He shall find the stubborn thistles bursting
All voluptuous garden roses."
Checks, and the Habit of Accuracy in
JOSEPH V. COLLINS, STEVENS POINT, wis.
GLENTY of experience convinces teachers that assur
ance plays a large part in individual development. Great strides can be made by a pupil in a short time if he has confidence in his powers, and feels that
he understands perfectly what he is doing. On the other hand, a pupil who is careless and inaccurate either in calculation or thinking comes to lack confidence in his powers, and may actually retrograde instead of progress. This parting of the ways, i. e., gaining confidence or losing it, occurs every time a new topic or a new subject is taken up. For this reason
. it is exceedingly important in arithmetic that the habit of accuracy in both thinking and mechanical work should be inculcated from the beginning of the training.
In the business and professional worlds accuracy in figures is a sine qua non. From the young boy who has just entered a grocery as a clerk to the profound mathematician, all are expected to get the correct result every time. To accomplish this everyone along the whole line exercises great care in performing each step of an operation, and then repeats it to check the result. Though twenty years have elapsed, the writer remembers as yesterday hearing the most eminent mathematician that has ever lived in this country say on one occasion: “I have just got a result which is extremely interesting to me, but I must go over my reasoning again to see that there is no mistake. I may have made some slip, and I shall not feel sure it is right till I go over it again.” The bookkeeper's accounts, however long, must balance to the cent, and this is accomplished by checking in one way or another, as by double entry or otherwise. The physician's or pharmacist's calculation of his doses must be right to give desired results, and so the calculation is repeated. The surveyor's long calculation of areas must be accurate, else he loses his reputation at once; hence he always checks up his calculation by totals of latitudes and departures. The expert accountant who prepares interest, logarithmic, or other tables must have every figure correct. The author of a set of mathematical tables recently issued offered one dollar a figure for every figure found wrong in a book full' of figures. It is only the schoolboy and often his teacher who sets up another standard, viz., that if at first you don't succeed in getting the answer in the book, try, try again.
One of the chief improvements in arithmetic asked for by the Committee of Ten was accuracy and celerity of calculation. To get this accuracy no plan was proposed other than the multiplication of simple concrete problems. The end proposed was good, very good, but it must be confessed that the means proposed were very general in character, and in themselves gave no assurance that the end would be accomplished.
Of the numerous text-books on arithmetic only a very few give attention to this very practical subject. One author brings into prominence the approximation method of checking. Another proposes to secure accuracy in mechanical calculation by keeping problems well within moderate limits as to difficulty, and then marking everything zero which has a single incorrect figure. Still -another early discusses the subject “ checks” and
” leaves the matter there. Now all this is needed and more. In some trigonometries a place and plan for checking are inserted in the model solutions. This course should be adopted in arithmetic also.
Some forms of problem admit of checking by a plan different from that used to get the result; others do not. In the latter case, the problem should be gone over again, once at least, verifying every figure. In the former case, this going over and verifying the problem should be performed first, and then, as a further safeguard, the test should be applied. The reason for this last course is apparent. If a mistake has been made which is found by the test, then the whole has to be gone over again. If, on the other hand, the going over the problem discovers the mistake, time is saved. Of course it is expected that common sense will be applied in all this. If the pupil has brought himself up to a good standard of accuracy, the going over the figures may be omitted, and only the test applied.
Let us now look into the different ways in which checking may be accomplished.
1. The most important of all is going over the work, this check being always applicable. It has, however, one important defect, viz., if a mistake is made once, there is danger of its recurrence.
2. Verification by approximation in round numbers. Thus if the cost of 2,487 lbs. of coal is asked for, find the cost of 1/4 tons. If the interest on $789.50 at 7 per cent for 2 yrs., 2 mos., 10 days is desired, find the interest on $800 at 7 per cent for 2 yrs., etc.
3. Verification of all the fundamental operations by casting out the nines. This test is very convenient and expeditious to apply, and should be known to all. In schools it is often explained but seldom applied.
4. Verification of operations by means of the reverse operation. Thus subtraction is proved by addition, division by multiplication, factoring by multiplying the factors together, etc.
5. Verification of problems by seeing whether the answer obtained satisfies the given conditions of the problem. This is the same as the verification of algebraic equations.
6. Verification by using a different method of solution. Thus the aliquot parts method is applicable to a great variety of problems, and can be used instead of direct multiplications and divisions or cancellation. Still other forms of verifying problems might be described, but perhaps the above will suffice. It is evident that to one who is willing to give time to the study of this question, there is more of practical value to be gotten out of it than might appear on the surface.
It is often contended that the teaching of mathematics has an ethical bearing. Suppose we assume that it has. The question arises, Do the too current methods of teaching arithmetic have a helpful or injurious effect on the morals of the pupil? The answer must be, They have the latter. The inculcation of the habit of taking care, on the other hand, is invaluable in other studies as well as arithmetic, and out of school as well as in school. Thus it may happen that bad arithmetic teaching may do much toward harming the lives of great numbers of pupils, whereas good arithmetic teaching may leave its mark in the character of the pupils and consequently on their lives.