feet high, when a space on the top about eighteen feet in diameter was leveled off and on this was built a fire hot enough to burn the ground red to a depth of four inches. Some time after this another foot of dirt was added to the top of the mound, where a like area on the top was leveled off and another fire built. Then it was finished by adding a foot-and-a-half more of dirt. The bottom layer of ashes has mixed with it many fragments of animal bones, fragments of pottery and implements. Several circular discs of mica were also found in it. In the top of the mound, above the upper fire-place, were found three skulls in different places; also, by themselves, lying side by side, two femora and between these a small copper ax. This ax was the first piece of copper found in this locality. The copper of which it was made probably came from Lake Superior, where the Indians worked large copper mines. The mica, too, must have come from a distance, as the nearest mica mines, worked by the Indians, were in North Carolina. Besides these, some interest is attached to the burnt layers. They are very similar in shape and position to those in mounds known to have been been built by the Cherokees, on the tops of which prisoners were tortured and burned, but the Cherokees were never known to have been nearer than Cincinnati.

Just across the river from this mound is the group on Mr. Payne's farm. This is on the level bottom just below the hill. There are four of these mounds and on the south and east sides of this group there is an earth embankment, now about six feet high and fifty feet wide, in the form of a semi-circle. On the southwestern side there is a gap, and at one side of this gap a small mound. The circle was probably part of a fortification, and this the entrance. Of the four mounds one is a large flat-topped mound about a hundred feet in diameter, the others are smaller with rounding | tops. On opening, two of the smaller mounds showed burnt layers similar to those in the Iceland mound except that they curved with the top of the mound instead of being flat. Nothing was found in these mounds whatever besides the burnt lavers.

Although ancient remains are so plentiful in this locality very little is known about them. They have never been systematically explored. And any map giving the location of mounds gives as the nearest, one near Troy and some near Evansville. This locality never having been explored, it is probable that one with plenty of time and money, especially the latter, would be amply rewarded in an exploration of the remains around here.

ROCKPORT, IND.

MATHEMATICS.

EDITED BY

ROBERT J. ALEY, Ph. D., Bloomington, Ind.

REPORT ON MATHEMATICAL EXAMINATIONS IN MICHIGAN.

A committee from the Mathematical Section of the Michigan State Teachers' Association was appointed in '96 "to consider the nature of the examinations to be set for teachers of mathematics.” This committee, through its chairman, Professor D. E. Smith of Ypsilanti, reported to the meeting of '97. The report is worthy the careful consideration of every one interested in mathematics. Some of the more important recommendations are as follows:

"The extent of the teacher's knowledge should greatly exceed the limits of an elementary course in mathematics, but great care is necessary in the setting of examinations to ascertain this knowledge. The extension of the teacher's knowledge beyond the subjects to be taught, should be in the way of serious mathematical subjects and not in the way of trivial problems or obsolete matter; e. g., while a teacher should understand true discount, to ask that a problem in this subject be solved would influence the teacher to give his pupils an erroneous notion of business."

The committee recommended that the following subjects in arithmetic be excluded from examination:

(a) Rules, formulæ, principles and most definitions.

(b) The naming of numbers above billions. (c) Roman numerals above two thousand. (d) Tables of denominate numbers not prac

tically used in science or in common business, or not needed for general information in reading and conversation. (e) Compound proportion. (f) Equation of payments.

(g) Partial payments, except by the method actually in use in this state.

(h) Compound interest beyond the simple case of computing such interest at rates ordinarily used in banks.

(i) Alligation.

Emphasis should be laid upon the following subjects:

(a) Pure arithmetic;

(1) The fundamental operations with integers and simple and decimal fractions. In time, contracted multiplication and division, so useful in solving physical problems, may be demanded.

(2) Elements of the theory of numbers.
(3) Square root, and in particular, the ex-
planation of the process.

(b) Applied arithmetic ;

(1) The problems actually in use in daily
life and in science.
(2) Especially the computation of simple
interest or bank discount upon a
non-interest-bearing note discount-
ed upon the day of its date, com-
mercial discount, and commissions
as usually reckoned to-day.

(3) The application of ratio and propor-
tion to common problems of science,
but not the ordinary text-book prob-
lems in these subjects.

(4) The common problems of mensuration.

(5) Compound numbers, including only work with two or three denominations in a single problem.

SPEER'S ARITHMETICS.

The statement of principles by Superintendent Speer in last month's EDUCATOR was both interesting and valuable. It may be worth while to examine his arithmetics. In the preface to the Primary Arithmetic, the author states quite fully the ideas that underlie the book. A few quotations may not be amiss. "The fundamental thing is to induce judgments of relative magnitude." "The one of mathematics is not an individual, separated from all else, but the union of two like impressions: the relation of two equal magnitudes." "It is not to be forgotten that there is a wide difference between seeing that the relation of two particular things is 8, and realizing 8 as a relation, realizing it in such a way that it can be freely used without misapplying it." "The condition of the child determines what he should do." "We may be so successful in training the child to reproduce as to destroy his power to produce." "Out of self-activity comes the self-control which gives strength to persist."

The introductory chapter upon "Mathematics" is interesting, but many mathematicians would not assent to all of the conclusions. It is urged that mathematical work should not be begun too early. "Premature attempts to initiate the pupil into the ideas of mathematics will bewilder him with the mechanism of the subject and create a condition unfavorable to the perception of mathematical or any other truth." No doubt many failures in arithmetic are due to premature beginning. Perhaps the real key to the primary work is found in this statement: "A living apprehension of the

fact that mathematics deals with definite relations of magnitude suggests the mode of beginning the study." "If this be accepted, then comparison with all that it implies follows as a necessary consequence." In summing up, Mr. Speer says: "Make definite relations the basis, and the integer and the fraction are each seen as a ratio; geometry, arithmetic and algebra merge insensibly into one another. With definite relations as the center, it becomes clear that if we would teach mathematics, and not the mere mechanism of the subject, we must look to the development of the representative and comparative powers. Only thus can we lift arithmetic from a matter of memory, routine and formula to its rightful place as a means of enlarging the mind." The work in the primary book consists very largely of the comparison of geometric solids and surfaces, with some introduction of weights and measures. The author has attempted to arrange the work so that the spontaneity of the child will be preserved. Frequent hints and directions to the teacher are inserted.

The elementary arithmetic continues the ideas of the primary. Many good things are said in the introduction, but one of the best is on "Time :" "The time in which a given amount of work should be done varies with the class and the individual. Work above the ability weakens; so does work below it. A fixed amount in a fixed time is fatal." It would be a great gain to education if the last thought could be burned into the consciousness of every teacher in our state. The book consists exclusively of exercises, most of which are in ratio. In the hands of a teacher thoroughly in sympathy with the Speer idea, these books would certainly give excellent results. To the teacher who does not wholly accept Speer's notions of mathematics, these books will offer many valuable suggestions.

THAT DIFFICULT PROBLEM AGAIN.

"Where shall a pole 120 feet high be broken that the top may rest on the ground 40 feet from the base?"

The arithmetical rules given in THE INLAND EDUCATOR for May, page 169, tell "how to get the answer" but convey no idea of the principle on. which they are based. The solutions given are algebraic with x and y omitted.

I submit the following: In a right-angle triangle, the square of the ratio of the base to the other two sides, as a whole, equals the difference of the other two sides as compared with their sum. In the above problem the ratio of the base to the other two sides is . The square of }=}. The difference of the other two sides being, one side equals, the other side. of 120=53}, the

It seems that some entertain the opinion that the above problem is not arithmetical.

I submit the following solution which I think is arithmetical:

A little investigation will prove the hypotenuse of a right-angle triangle is of the perimeter, and that the two adjacent perimeter.

sides are of the

The perimeter of the above problem is 120 feet, +40 feet 160 feet.

of the perimeter the two sides.

1 of the sides is 40 feet, or of the perimeter.

Then, -- of the perimeter which is the other side.

of 160-531 the height of the stump. Again, 40 feet+120 feet 160 feet which is the perimeter.

r of perimeter the two sides.

of perimeter the hypotenuse.

40 feet or of perimeter is one of the sides. Then, -1}, and of 160 feet-533 feet, the other side and the required result.

As I have never seen a similar solution to such problems I would be pleased to hear from others. E. E. ELLIS.

FARMERSVILLE, IND.

A THIRD SOLUTION.

Here is that two-cent stamp problem again. It was only accidentally that we solved it. "We" means teacher and pupils. The latter had just worked the problem "To find any side of a rightangle triangle when the other two sides are given," Art. 437, Complete Arithmetic, when I saw the "A Difficult Problem" in THE INLAND EDUCATOR for March. The remarks connected with it induced me to give it to the class for solution. The first result of the discussion of the problem was this: "Given the base and the sum of the other two sides (A+H) of a right-angle triangle, find the altitude." On the blackboard was drawn a right angle triangle with the square of each side, the sides being 3, 4, 5, respectively. This triangle was used as the basis of operation.

It was seen that A+H=9, B=3 or 3(A+H), A= B. In the other triangle A+H=120, B=40, or (A+H), consequently the relation of the three sides is the same in both triangles. As in the greater triangle B=40, A=‡ of 40, which is 53}. Or, 3=40; 1=13}; 4=53}; 5—663; 531+663=120= A+H.

The same triangle may be so constructed that the base is 4, and the altitude 3. Then B must be (A+H). The problem altered accordingly would then read, "Where shall the pole be broken that the top rests on the ground 60 feet from the base of the stump?"

Solution: 4-60, 3=45 (A), 5=75 (H). 45, ans. In the same manner the height of the tree can be found when the base and either of the other two sides are given.

With one or two exceptions, examples in our (Indiana) arithmetic can be solved without squaring two sides and extracting the square root. THEO. DINGLEDEY.

NOTES.

Dr. David Eugene Smith, professor of mathematics in the Michigan State Normal School, will be succeeded at the end of the present year by Professor E. A. Lyman from the mathematical department of Michigan University. Dr. Smith goes to the principalship of the Brockport (N. Y.) Normal.

Mr. John C. Stone, an able and enthusiastic teacher of mathematics, leaves the Elgin, Illinois, High School to become head master of mathematics in the Lake Forest Academy.

A. E. Winship in a recent number of the Journal of Education says: "There is an attempt in some quarters to minimize arithmetic. I think it is to be magnified. I agree fully with President Eliot that too much time has been given to the subject, that too much time is still given to it in some schools, but the greater trouble is that the children know so little about it, after all the time that it receives. . . . Whatever may have been true of the tendency to over-emphasize arithmetic in the past, there are many schools that practically ignore it now. In some places the schools are losing the respect of the business community because boys, especially, leave school with so little power or training in numbers."

Professors Fisher and Schwatt of the University of Pennsylvania have in press a school algebra. They are both well-trained and brilliant mathematicians. They have given to the preparation of this book their best thought for a number of years

They have had as their aim the simplifying of algebra without losing its mathematical character. The careful reading of the first few chapters shows that they have been successful. A full review will be given as soon as the book comes from the press.

Scott, Foresman & Co. have recently issued a School Geometry by J. F. Smith, of Iowa College Academy. The book differs materially from the usual text on this subject. The first part is an excellent introduction from the concrete side. This is followed by the more formal part in which nearly all of the ordinary plane geometry and a considerable part of the solid is developed. The subject is approached gradually and in a manner that will surely interest the pupil. The book might serve a better purpose if it were divided and bound in two volumes. The concrete part with a few additions would make a splendid supplementary work for seventh and eighth grade pupils, and would be a good preparation for formal geometry. The remaindert of the book, with some changes and additions, would be a good highschool text. The teacher of geometry will find the book a very suggestive one.

In John Kersey's edition of Wingate's Arithmetic, 1671, the following discussion of Decimals, then a new subject, is found: "It is hard to determine who was the first that brought Decimal Arithmetick to light, though it be a late invention, but without doubt it hath received much improvement within the compass of a few years by the industry of Artists, and now seems to be arrived at perfection. The excellency thereof is best known to such as can apply it to the practical part of the mathematicks and to the construction of Tables, which depend upon standing or constant proportions, such as Trigonometrical Canons, Tables for computing of Compound Interest, etc., in which cases decimal operations do afford so great help that (in my opinion) many ages have not produced a more useful invention; but it may be objected that decimal arithmetick for the most part gives an imperfect solution to a question, this I grant, yet the answer so given may be as useful as that which is exactly true; for in common affairs, the loss of obs part of a grain or of an inch, etc., to wit, any quantity which can not be seen, is inconsiderable; but I would not be mistaken, for in extolling Decimals I do not cry down Vulgar Fractions, since experience sheweth that Decimal Fractions are commonly abused by being applyed to all manner of questions about money, weight, etc., when indeed many questions may be resolved with much more facility by Vulgar Arithmetick. [Here follows an example.] I might instance the like inconvenience divers ways were it not for loss

of time; so that the right use of decimals depends upon the discretion of the artist." The last statement is especially good.

PSYCHOLOGY OF NUMBER.

Number is a function of intelligence; hence it is not a characteristic of external material objects. As attention is not a distinct faculty of the mind, but a condition of all other fundamental psychical activity, so the number concept stands for an activity that is associated with all rational energy. While number may be occasioned by material objects, it is not an attribute of them. Number is an instrument imposed by the mind upon external objects, and is a chief agency in the classification, organization and relating of knowledge. Subjectivity and psychical nature are the most palpable attributes of number.

Mental activity tends constantly to project itself into and upon external objects. That the tendency of thought is to manifest itself is an old saying of philosophers. The resulting reaction gives rise to quantitative and qualitative ideas.

Through a process of sense-perception the ego. becomes conscious of the non-ego, or of externality. The mind, animating an extended sensorium, is made aware of extension as an attribute of the body. When the body which is itself extended, moves, it comes into contact with other objects, and consequently realizes a material limit. By a process of transference the mind recognizes a mental limit as well as a material one. The idea of limit, which is an outgrowth of resistance encountered, leads naturally to estimation, and to efforts at balancing mental forces against outward obstructions.

As the struggle to overcome nature becomes keener, man realizes the necessity for husbanding resources and exactly adjusting internal energy against external opposition. This gives rise to the equation which is fundamental in psychical activity. During this process all of the mind has been active. Synthesis is preliminary in soul activity; every mental act may be regarded as made up or integrated of many others.

Analysis has always seemed to me to be an inferior form of activity. The nature of the mind is to be a whole, to make wholes and to rest in wholes. Inductive reasoning and inductive teaching have almost assumed the proportions of an educational fetich. I have never been able to see why the mind, even in so-called inductive reasoning, does not begin with a vague and somewhat shadowy hypothesis. The mind then tries to find the general proposition in different particulars. It has always seemed to me that induction, as usu-