MECHANICAL CONSTRUCTION OF THE AREA OF A

CIRCLE.

By CHAUNCEY SMITH, of the Suffolk Bar, Boston.

PROPOSITION. The area of a circle is equal to the rectangle contained by its semicircumference and semidiameter.

Children usually find this proposition in their arithmetics, in the form of a rule, before they have any knowledge of geometry; and I therefore propose the following simple, graphic method of demonstration.

Cut a circle from a piece of card, and bisect it as in Fig. 1. Divide each half into an equal number of sections, cutting from the centre, C, but leaving them all joined together at the circumference. Next,

*This graphic method may be exhibited equally well upon the blackboard, by drawing the figures, and showing how one is made from the other; and it will be found of great advantage in those cases in which the pupil is called upon to work by an arbitrary rule, as in most of the arithmetics. To the same end, the rectangle may be divided into squares, and the pupil made to see that if one of these squares is taken as the unit of surface, that the surface of the whole rectangle will be measured by the product of its base by its altitude; that is, this product will show how many such units the rectangle contains. Indeed, the 2

VOL. II.

in Fig. 3. If the number of sections is very great, ADB and AEB

altitude are the semi-circumference and semidiameter of the circle, and their product is the measure of its area, which is therefore the measure of the area of the circle,

ON THE PRIME SEVENTH AS AN ESSENTIAL ELEMENT IN THE MUSICAL SYSTEM.

By HENRY WARD POOLE, Engineer, Boston.

It is now ten years, since, by original investigation in the mathematical, mechanical, and practical departments of music, I was led to the belief that this science has a solid foundation in the relations of numbers, and that all the supposed impossibility of Just Intonation and the necessity of Temperament, have their origin only in the short-sightedness of the theorist, and the unskilfulness of the practitioner.

Having settled upon the rule that musical ratios must not exceed a certain limit of simplicity (the limit to be determined by the

areas of all the surfaces investigated in plane geometry may be exhibited to the eye by simple construction; and quite young children may readily be taught to understand as well as make the constructions for themselves. - ED.

Since receiving the above, the same construction has been sent us by R. C. MATTHEWSON, Esq., of San Francisco, California; and we also find it in LUND'S Geometry, an English book of recent date.

ability of the ear to appreciate them), it was stated that those ratios only were admissible which were derived from the prime numbers 2, 3, 5, and 7. That the three lower primes 2, 3, and 5 belong to the musical system has been universally admitted; but no one, before myself, so far as I know, has made this claim for the prime seven.+

* American Journal of Science, Second Series, Vol. IX. pp. 68, 199.

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† I do not wish to conceal the fact, that even now the principle of just intonation (or the possibility, in theory or practice, of exact fifths, thirds, &c.) is denied by high mathematical authorities. Sir J. F. W. HERSCHEL, in his treatise on Sound, declares, that " singers, violin players, and all others who can pass through every gradation of tone, must all temper, or they could never keep in tune with each other or themselves." [The work of HERSCHEL not being at hand, this extract is copied from the treatise on Sound, by Professor BENJAMIN PEIRCE, of Cambridge, who has reproduced (with his indorsement, it is presumed,) these and like views of HERSCHEL] By a late letter from Sir JOHN HERSCHEL, dated Collingwood, June 14th, 1859, addressed to the Musical Pitch Committee, at the Society of Arts, he evinces his continued belief in Temperament as inherent in music, and his opinion that this temperament gives some peculiar character to the different signatures or keys in music in general. He says, in regard to the concert pitch :

"All are desirous that when once lowered, it should be kept from rising [1] again, to which there is a continual tendency arising from a distinct natural cause inherent in the nature of harmony; namely, the excess (amounting to about eleven vibrations in ten thousand) of a perfect fifth over seven-twelfths of an octave, which has to be constantly contended against in upward modulations, whenever violins or voices are not kept in check by fixed instruments. But perhaps all are not aware that the evil of fine ancient compositions having thus been rendered impracticable to singers in their original normal key involves the sacrifice of the adaptation of the peculiar character of the key (a character intended and felt by the composer), and the substitution of a totally different incidence of the temperament [2] in the series of notes in the scale, and goes therefore to mar the intended effect, and injure the composition, as much as an ill-chosen tone of varnish would damage the effect of a fine Titian."

1. There is nothing better to test the "natural tendency" in this respect than a good gleeclub without accompaniment. If a high pitch is taken and they are fatigued, the pitch will gradually fall. If they start with too low a key-note and are in good spirits, the tendency will be to rise to the better pitch. It does not appear that temperament affects the concert pitch.

2. Observe the same glee-singers. They sing in every key with the same relative intervals, and do not use a "different incidence of temperament,” in different keys. Did any composer of glees wish such temperament? If so he should indorse his score something after this manner: "Four flats, equal temperament" (as the composers of fugues for the organ have actually done;) or "Four flats, with a great wolf in A flat, and a whelp in E flat."

I only desire here to put on record for historical reference the most respectable authorities of this day against Just Intonation, and to prove that the views I put forth have such opponents, and hence need to be told.

The interval 4: 7 derived from the prime seventh has not been unnoticed, as a curiosity in acoustics; and it is occasionally referred to as the "Za" of Tartini. A living writer, whose statements are entitled to the highest respect, and whose works contain most able arguments in favor of Just Intonation, says of the sounds produced from the prime seventh: "They may be called anomalous. They are wheels, but not wheels which will fit in with the previously constructed parts of the machine, and therefore they are left on one side."

The sound 4:7 has been known to be the seventh harmonic of the horn and æolian string, but has been called a "false" note, and has been rejected even by the advocates of just intonation, as opening the way for inextricable complication in theory and practice. It will from this, appear necessary to make the declaration which is the subject of this paper, and which is as follows:

The Prime Seventh belongs to the Musical System; its ratios are altogether appreciable by the common ear, and are in constant use in common music. It is this which constitutes, when added to the common chord, the concord (falsely called the discord) of the Seventh, and this element, combined with the other prime chords of Octave, Fifth, and Major-Third, makes the great variety of noble harmonies in which cultivated and uncultivated ears delight.

The prime seventh is necessary to complete the series of simple ratios, which extend as far as 10; and it was by noticing the blanks which its omission would leave that its necessity was discovered. The series is as follows:

1:2, 2:3, 3:4, 4:5, 5:6, [6: 7, 7:8,] 8:9, 9:10

London, 1857.

6 vols. Both are in the

*Gen. T. PERRONET THOMPSON. Just Intonation. p. 72. 2d Edition. See also his "Exercises, Political and Others." London. 1843. Boston Athenæum.

or, if written as below, we shall have the natural series of harmonics, or what may be called the primary or

HARMONIC SCALE.

1:2:3:4:5:6:7:8:9:10.

As some reason should be assigned to the mathematician for not extending the series by the introduction of the Prime Eleventh, it will be found in the inability of the human ear to appreciate such complicated relations. The "Chord of the Eleventh" exists in nature, and I am able to tune it and to recognize its harmony in combinations specially made for the experiment; yet, so far as my examination of the works of the masters has extended, it has not been used by them in their written music, and perhaps never, unless possibly in the harmonics of a Paganini. I do not claim for it a place in our practical system of music, but leave it where all the former theorists have set the Prime Seventh. The Eleventh may hereafter be admitted, when the musical faculties of men have been sharpened by familiarity with the more simple concords in their purity, and when music is carried to a higher degree of refinement.

From the Harmonic Scale may be derived, by combination of its chords, an indefinite variety of other scales. The Octave is divided into eight intervals, which are convenient for melodic use, and the result is popularly called the Diatonic Scale, which, although generally taken as the basis, in explaining music, is not a primary, but a secondary Scale. The method of forming it, according to all former treatises, is, to take common chords (4:5:6) upon the tonic, dominant, and subdominant. Thus, the scale of C is tuned by taking a Fifth and Major-Third on C, on G, and on F, and bringing all the notes within the same octave.

But the introduction of the prime seventh allows of another division, in which only two fundamentals are employed; namely, the