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will soon come to take me to be where he is. Yea, Lord Jesus, I thank thee, thou beginner and finisher of my faith, who hast brought me, a foolish wanderer, straying a thousand ways from the direction of my journey, diverted and delayed in a thousand by-occupations, so far that now I see before me the bounds of the promised land, and have only to cross the Jordan of death, to attain even unto thy loveliness. I praise and glorify thy holy wisdom, O my Saviour, that thou hast given me on this earth no home; but that it has been for me only a place of banishment and pilgrimage; and I can say with David, 'I am thy pilgrim and thy citizen. I can not say, like Jacob, “My days are few, and they attain not unto the days of my fathers,' for thou hast caused it to come to pass that they surpass

the days of my father and my grandfather, and many thousands who have passed with me through the desert of this life. Why thou hast done this, thou knowest. I commit myself into thine hands. Thou hast always sent an angel unto me, as unto Elias in the desert, with a morsel of bread and a draught of water, that I should not die of hunger and thirst. Thou has preserved me from the universal foolishness of men, who always mistake pleasure for real good; the road for the destination ; striving after rest; the inn for a home; and pilgrimage for their country; but me hast thou led, and even forced, to thy Horeb. Blessed by thy holy name !"

PEDAGOGICAL WORKS OF COMENIUS. 1. JANUA LINGUARUM RESERATA AUREA

SEMINARIUM LINGUARUM SCIENTIARUM OMNIUM, hoc est, compendiosa Latinam (et quamlibet aliam) linguam, una cum scientiarum artium que omnium fundamentis, perdiscendi methodus, sub titulis centum, periodis mille comprehensa. Editio postrema, prioribus castigatior et mille circiter vocabulis auctior, cum versione Germanica et Gallica, absolutissimoque titulorum et vocum indice. Amstelodami apud Joannem Janssonium. 1642.

I am not acquainted with the first edition. Comenius' preface is signed with " Scribebam in exilio 4 Martii. 1631."

2. PhysicAE AD LUMEN DIVINUM REFORMATAE SYNOPSIS. Lipsiae, 1633.

3. ORBIS SENSUALIUM Pictus, hoc est omnium fundamentalium in mundo rerum et in vita actionum, pictura et nomenclatura. Editio secunda, multo emaoulatior et emendatior. Noribergae typis et sumptibus Michaelis Endteri, 1659. The visible world ; that is, the representation and names of all the principal things of the world and occupations of life.

I am unacquainted with the first edition. Of the later ones, I have an Orbis Pictus Quadrilinguis, in Latin, German, Italian, and French, which was edited by Coutelle and published by Endter, in 1755.

4. Opera DIDACTICA Omnia, variis hucusque occasionibus scripta, diversis que locis edita, nunc autem non tantum in unum, ut simul sint, collecta, sed et ultimo conatu in systema unum mechanice constructum, redacta. Amsterdami impensis D. Laurentii de Geer excuderunt Christophorus Conradus et Gabriel a Roy. Anno, 1657. 4 vols., folio.

Volume I. contains the following, written between 1627 and 1642: 1. De primis occasionibus quibus huc studiorum delatus fuit author, brevissima relatio.

2. Didactica Magna. Omnes omnia docendi artificia exhibens. 3. Schola materni gremi, sive de provida juventutis primo sexennio educatione.

SIVE

ET

4. Scholae vernaculae delineatio. 5. Janua Latinae linguae primum edita. (The first edition of the Janua.) 6. Vestibulum ei praestructa. 7. Proplasma templi Latinitatis Dav. Vechneri. 8. De sermonis Latini studio, 9. Prodromus Pansophiae. 10. Variorum de eo censurae, &c.

Volume II. contains treatises written from 1642 to 1650; especially those of his Swedish engagement, viz. :

1. De novis didactica studia continuandi occasionibus.
2. Methodus linguarum novissima.
3. Latinae linguae vestibulum, rerum et linguae cardines exhibens.
4. Januae linguarum novissimae clavis, grammatica Latino-vernacula.

Volume III. contains treatises written by Comenius in Hungary, from 1650 to 1654, viz. :

1. De vocatione in Hungariam relatio.
2. Scholae pansophicae delineatio.
3. De repertis studii pansophici obicibus.
4. De ingeniorum cultura.
5. De ingenia colendi primario instrumento, libris.

6. De reperta ad authores Latinos prompte legendos et clare intelligendos facili, brevi, amoenaque via.

7. Eruditionis scholasticae pars 1. Vestibulum, rerum et linguae fundamenta ponens.

8. Eruditionis scholasticae pars II. Janua rerum et linguarum structuram externam exhibens. This includes

Q. Lexicon januale.
b. Grammatica janualis.

c. Janualis rerum et verborum contextus, historiolam rerum continens. This is a revision of the Janua reserata, in one hundred chapters and one thousand paragraphs, as in the first edition.

9. Eruditiones scholasticae pars III. Atrium, rerum et linguarum ornamenta exhibens. This is, like the Janua, in one hundred chapters and one thousand paragraphs, but one grade above it.

10. Fortius redivivus, sive de pellenda scholis ignavia.
11. Praecepta morum in usum juventutis collecta. Anno 1653.
12. Leges bene ordinatae scholae.
13. Orbis Pictus. Merely a sort of announcement of the work.

14. Schola ludus ; hoc est, Januae linguarum praxis comica. This is, substantially the contents of the Janua linguarum in the form of a dialogue.

15. Laborum scholasticorum in Hungaria obitorum coronis. An educational address delivered at his departure from Patak, in 1654.

Volume IV. includes the treatises written by Comenius in Amsterdam, up to the year 1657, viz. :

1. Vita gyrus, sive de occasionibus vitae et quibus autorem in Belgium deferri, iterumque ad intermissa didactica studia redire contigit.

2. Parvulis parvulis, omnibus omnia, hoc est, Vestibuli Latinae linguae auctarium, voces primitivas in sententiolas redigens.

3. Apologia pro Latinitate Januae linguarum.
4. Ventilabrum sapientiae, sive sapienter sua retractandi ars.

5. E scholasticis labyrinthus exitus in planum, sive machina didactica mechanice constructa.

6. Latium redivivum, hoc est, de forma erigendi Latinissimi collegii, seu novae Romanae civitatulae, ubi Latina lingua usu et consuetudine addiscatur.

7. Typographeum vivum, hoc est; arscom pendiose et tamen copiose ac eleganter sapientiam non chartis sed ingeniis imprimendi.

8. Paradisus juventuti Christianae reducendus, sive optimus scholarum status, ad primae paradisiacae scholae ideam delineatus.

9. Traditio lampadis, hoc est studiorum sapientiae Christianaeque juventutis et scholarum, Deo ei hominibus devota commendatio.

10. Paralipomena didactica. It may be added, that Comenius revised an edition which appeared in 1661, of the Theologia naturalis sive liber creaturarum of Raymundus de Sabunde.

XVII. EDUCATIONAL MISCELLANY AND INTELLIGENCE.

ON THE EFFECTS OF INITIAL GYRATORY VELOCITIES, AND OF RETARDING FORCES,

ON THE MOTION OF THE GYROSCOPE,

BY MAJOR J. G. BARNARD, A. M

Corps of Engineers, U. S. A.

In one of the concluding paragraphs of my first paper on the Gyroscope (Am. Journal of Education, June, 1857,) I stated that “ an initial impulse may be applied to the rotating disk in such a way that the horizontal motion shall be absolutely without undulation. An initial angular velocity such as would make its corresponding deflective force equal to the component of gravity g sin 0, would cause a horizontal motion without undulation."

The statement contained in the last sentence quoted, is not rigidly true; for besides the component of gravity, there is another force to be considered, viz., the centrifugal force due to the gyratory velocity, which acts either in conjunction with, or in opposition to the component of gravity, according as the axis of the disk is above or below a horizontal.

In this last position this force is null (as regards its effects in sustaining or depressing the axis), and to this angular elevation of the axis the statement quoted is true without qualification. The assumption of an initial horizontal velocity requires only a new determination of constants for equations (a) and (c) (pp. 541, 542, June No.). If we make, in those equations

0=«, p=90°, y=90°, a=-sin a, Vi=m, vy = 0, vzan, (in which m is the assumed initial velocity) and determine the constants ħ and I therefrom, the equations of motion will become

Сп sin 20

(cos 0 - cos a) + m sin a
dt A

(1)
d02
sin20 +

m2
d12
d t2

A and from them we get

d02
2 M 97

2 Cm m C2 n2
sin 20
sin 20..

-(cos - cosa)
d 12
A

A

A2

dy

}

dyz

[ocr errors]

=

sina

m2 (cos 0 + cos a)](cos 6 – cosa) (2)

do

dy From this we get =0 when cos 0 cosa =0; and as

is not zero dt

dt for this initial elevation, it indicates, instead of a cusp, a tangency to the horizontal here.

This paper is intended to give a more rigidly mathematical demonstration of the effects of “ retarding forces” than is given in (December No. p. 529,) of this Jourpal; and to give the theory of the “motions" of the Gyroscope a more general form, by the introduction of Initial Gyratory Velocities."

If the curve described is horizontal without undulation, the other factor of the second member of eq. (2) should likewise become zero with 0=a: an effect which may ensue from a suitable value given to m. The value of the deflecting force due to a given angular velocity m is

с (p. 552, June number) mn, and if we suppose this equal to the com

[ocr errors]

MgY

sin a. Сп

ponent of gravity g sin a, we shall have m =

If we substitute this value of m in the second member of equation (2) and assume a = 90° the factor in question becomes zero for 0=a, and the maximum and minimum values of 0 are the same, indicating a horizontal motion without undulation.

For every other initial elevation than 90° a different value of m is required to produce this result, in consequence of the influence of the centrifugal force of gyration at other elevations. With a=90°, equation (2) becomes 2 M9Y

2 Cm m

C2 n2
sin2
-sin 2 0 -

cos 0 -ma cos 0 0
d 12
A

A

A2 Placing the first factor of the second member equal to zero and solving with reference to cos 0 we get (recollecting the value given to ß in our former article) A m2

Am2

Cmn cos 0 = -82

(4) 4 MgY

*

[ocr errors]

[

2

+

82 +

4 M9Y

+1-M91

[merged small][ocr errors][merged small][ocr errors][ocr errors][merged small][ocr errors][ocr errors][ocr errors]
[ocr errors]

For m=0, equation (3) expresses the cycloidal curve with cusps a, a', a", &c., as has been already shown in our former investigation. For mo but < the minimum value of 0 derived from equation (4) is

Сп greater than when m is zero, while instead of a cusp (there is as has already been observed) a tangency at a, and the curve has the wave form ab, a'b', (the points b,b;' b;', &c. being higher than 6 b' b'').*

the curve unites with the horizontal aa'a'a'' and

Сп there is no undulation; equation (4) giving cos 0 = 0, or 0= 90°.

Mgy When m=

* In reality, the amplitudes, a a', a'a'', of the undulations become increased, at the bame time that the sagittæ are diminished, but, for the sake of comparison, I have represented them the same for each variety of curve.

and a

When m
Mgy do

becomes still zero with 0=a=90°; but this

Cndt instead of a maximum is now a minimum value of 0, for the value of 0 which satisfies equation (4) is greater than 90°, and the curve ab, a' ba', &c., undulates above the plane aa'a". 2 M9Y

1 Finally when m= equation (4) will give cos 0 =Сп

282 substitution of this in the first equation (1) (making a= 90°), will give dy

-=0: showing that the curve makes cusps at its superior culminations, dt. and that the common cycloidal motion is resumed. In fact the value of

if (p. 547, June number) at the lowest point b of the cycloid, is, dt

BNT (substituting the values of ß and ) exactly equal to

Сп value of the sagitta u corresponding to eb is what we have just found for

viz.

2 If now, retaining m constant at this value to which we have brought it, we increase the rotary velocity, n, or vice versa, a curve with loops, (fig. 2,) may be described, as it can be shown that, for the maximum value

becomes negative.*

dy

1

2 MIY, and the

1

of 0,

dy dt

[blocks in formation]

In my supplementary paper in the December number of this Journal I have endeavored to show how the theoretical cycloidal motion of a simple solid of revolution is modified by the retarding forces of friction and the resistance of the air, and that the theory explains all the phenomena observed in the ordinary gyroscope.

It may be objected however that the nature of the curve given in Fig. 1, (p. 531,) is in some degree assumed, and I therefore wish to show that it can be confirmed by mathematical demonstration.

The rotary velocity n of the disk is supposed to be gradually destroyed through the retarding forces of friction at the extremities of the axle, and of the resistance of the air at the surface.

Without attempting to give analytical expressions for the retarding forces, it is sufficient to say that the rotary velocity, at the end of any

*If m is made negative and small (i. e., a backward initial velocity given) a looped curve like the above, but lying below the plane a a' a", results. All these curves (n being always supposed very great) are but the different forms of the "cycloid” known as prolate, common, and curtate cycloids; the common-a particular case of the curve-corresponding to the particular case of the problem in which the initial

2 Mgy gyratory velocity is either zero or has the particular value

Сп

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