there are 3 tens in 35. For 47, we must set down 7 and carry 4, because there are 4 tens in 47." These, and similar observations, should be made to arithmetical classes whenever time will admit. But we will attend to the calculation again. The second boy, No. 2, in the semicircle, says, "2 that I carry, and 1 is 3, and 5 are 8, and 4 are 12, and 3 are 15, and 2 are 17, and 5 are 22, and 2 are 24; set down 4 and carry 2 to the next, because there are twice 10 in 24." The residue of the class take their turns in adding up a column; but as No. 4 adds up the last column, No. 5 must begin with the first column, in the manner that No. 1 began the same; then No. 6 will add the second column. No. 7 the third, No. 8 the fourth, &c. in rotation till they have had an equal share in exercising their faculties.

22. When a new task becomes expedient, deface one or two figures in each column with a piece of sponge, or cloth moistened in water, and replenish those vacancies with different figures. If we perceive a readiness or expertness in a majority of the class when calculating, then it may be proper to fill those vacancies with higher numbers; but here let it be repeated, that, such steps or enlargements must be made with caution; without advancing too far at a time; lest we embarrass the young minds, retard their progress, and create a discouragement.

23. This last mentioned evil has been prevalent from time immemorial; but that era is fast approaching, when the whole circle of science will be divested of puzzling, mysterious, and partial schemes.

24. In the next place we will suppose an older class are called up to the board, and that they are as ready in say

ing "8 and 7 are 15, and 9 are 24, and 7 are 31;"&c. as the others were in saying " 4 and 2 are 6, and 3 are 9, and 2 are 1. In this case, let the lesson on the board consist of more columns, and of more and higher figures in each column. But in all cases beware of extravagance, that time may not be taken up in unuseful projects; that is, in calculating such hard or uncommon lessons, which seldom or never happen in ordinary practice. Usefulness and despatch are objects aimed at by all advocates for Lancaster's method of education.

AN ADDITIONAL METHOD.

25. There is an improvement in teaching arithmetic on Lancaster's plan, and for the satisfaction of readers, a lesson is hereunto subjoined.

26. I submit this mode of operation to the judgment of Teachers, and will let them make use of it as they think proper in a few of the first lessons.

The first six cards of this book will nearly supersede the necessity of resorting to this mode of calculation, and clause 18 will show how to determine when a scholar may be advanced to a different degree of lessons. Another

alteration different from Lancaster's mode is this, add immediately whatever there be to carry to the next column, instead of reserving it till you have added

27. THE PESTALLOZZIAN PLAN.

I will mention this plan for exercising the mind on calcu lation, and then proceed to the government of a school. The seven subjoined periods form a characteristic extract from a book entitled, "Sketch of a Plan and Method of Education, founded on an Analysis of the human Faculties, and Natural Reason, suitable for The offspring of a Free People, and for all Rational Beings. By JOSEPH NEEF, formerly a coadjutor of Pestallozzi, at his school near Berne, in Swisserland.

Philadelphia: Printed for the Author, 1808."

The characteristic periods are as follow:

"There lives in Europe, beneath the foot of the Alps, an old man, whose name is PESTALLOZZI; a man as respectable for the goodness of his heart, as for the soundness of his head. This man, endowed by nature, or rather nature's God, with the felicity of an observing mind, was forcibly struck with the vices, follies, and extravagancies of the superior ranks, and the ignorance, superstition, and debasement of the inferior ranks of society. He perceived that from these impure sources flowed all the miseries that afflicted his unhappy fellow-creatures. Being no disciple of Zeno, the woes of his brethren naturally imparted their anguish to his sensible heart.

"The host of calamities under which he saw his fellowmen groaning, deeply grieved his feeling soul, and the gulph of evils into which he viewed mankind plunged, called forth the most cordial and sincere compassion.

Tears fell from his mourning eye, but they were manly tears. Far from being disheartened by such a sad spectacle, he had the courage to enquire into the causes of human misery; he went even a step farther, and endeavoured to find out an* wholesome remedy, calculated to destroy at their very source, those evils which inundated the world.”

This is the Patriot and Philanthropist, who invented an advantageous System of Education, to be pursued by every class, from the novice to the profound scholar.

I will give a hint of this method, and that in calculation only.

MODE OF OPERATION.

Instead of figures, let boys make use of peas, beans, kernels of Indian corn, short pieces of sticks, leaves of grass, of weeds, or any other convenient materials. To a certain number of these objects, add another number, then count how many there are in the whole: call this the sum total. From another number of these objects, take away a certain number, and ascertain how many are left: call this number which is left, a remainder. Thus we can perform Addition and Subtraction with real objects, instead of those which are imaginary.

This exercise will afford children pastime when they are out of school; it will keep their minds intent on the business they are sent to perform, and will not disappoint their guardians, nor grieve their hard-labouring and frequently indigent parents.

But these two rules are not all that can be performed by moveable objects without figures. Multiplication and

* When his sounded, a, is preferable to an.

Division, may be wrought with the same materials on a checker-board about two feet square; and this may be done to a considerable greater degree of profit, than is commonly received from the operations played on that fascinating time-waster.

28. By this board we can solve the following problems, and perhaps thousands more.

How many times one will two times two make? Two times two are four, and four times one are four. Answer, four times.

29. How many times one can we make of three times two? Three times two make six, and there are six times one in six. Answer, six times.