Herman, Jacob

Jacob German was born in the city of Basel on July 16, 1678 ^[5] . He studied at the University of Basel and graduated in 1696; a student of Jacob Bernoulli , under whose leadership Herman studied mathematics ^[4] . Initially, he hoped to engage in theology and in 1701 even took the dignity, but his penchant for studying mathematics won ^[6] . In his first work ^[7] , which was published in 1700 and aimed at refuting the attacks of the Dutch mathematician and philosopher on differential calculus , he attracted the attention of G. V. Leibniz , on whose representation German was elected a member of the newly established Berlin Academy of Sciences ( 1701 ) ^[8] .

Actively engaged in mathematics, Herman publishes a number of articles in the German scientific journal Acta Eruditorum , two of which ^{[9] [10]} attracted the attention of the most prominent mathematicians of the time ^[8] ; as a result, German, on the recommendation of Leibniz , was invited in 1707 to occupy the Department of Mathematics at the University of Padua . During his work in Padua (1707-1713), German gained great respect among Italian scientists and in 1708 was elected to the Bologna Academy of Sciences. Since 1713, Herman is a professor at the University of Frankfurt an der Oder ^{[4] [11]} .

In 1723, L. L. Blumentrost, in fulfillment of Peter I ’s intention to establish an academy of sciences in Russia, turned to the well-known German scientist H. Wolf with a request to recommend several European scientists for the newly established academy; Among the candidates proposed by Wolf was Herman. The last one agreed with Blumentrost’s letter, and on January 8 ( January 21 ), 1725, he signed a contract for five years with the Russian diplomat specially visiting Frankfurt-on-Oder , A.Golovkin , for his membership in the Academy as a professor of mathematics. German was the first of foreign scientists to assume the duties of a member of the St. Petersburg Academy of Sciences , for which he was called professor primarius 'first professor' (in other words ^[12] - “first academician”) ^[13] .

German arrived in St. Petersburg on July 31 ( August 11 ), 1725 . August 15 ( August 26 ), and he - among the first academics who arrived in the Russian capital - was introduced to Catherine I in her Summer Palace; at the same time, he delivered a welcome speech addressed to the empress, well received by all those present. It was Hermann who opened on November 2 ( November 13 ), 1725, the first meeting of the Petersburg Academy of Sciences (held before its official opening) and read the text of his article “De figura telluris sphaeroide cujus axis minor sita intra polos and Newtono in Principiis philosophiae mathematicis synthetice demonstratam analytica methodo deduxit ” , which analyzed the theory of the Earth’s figure proposed by Newton , according to which the Earth is a spheroid flattened at the poles ^[14] . This statement of Herman, by the way, provoked the objections of another academician - G. B. Bilfinger , who adhered to Cartesian mechanics and did not accept Newton's theory of gravitation ^[15] .

In the Petersburg period of his life, Herman worked intensively; About a dozen of his articles on mathematics and mechanics were published in the scientific journal of the Petersburg Academy of Sciences Commentarii Academiae Imperialis Scientiarum Petropolitanae . In particular, it was with an article by Herman entitled “De mensura virium corporum” ^{[16] that} opens the first volume of this journal (prepared in 1726, but published in 1728) ^[17] . When L. Euler , who also became an academician of the Petersburg Academy of Sciences, arrived in St. Petersburg on May 24 ( June 4 ), 1727 , then German, being his fellow countryman and distant relative (Euler’s mother was brought in by her second cousin ^[3] ), gave Euler all kinds of protection ^{[ 18]} .

In 1728, however, serious friction began between a number of academicians (including German) and the secretary of the Petersburg Academy of Sciences Johann-Daniel Schumacher ; the political situation in Russia has become more complicated. Under these conditions, German did not renew his contract (the term of which expired in 1730) and in September 1730 he was dismissed from the academy (with the title of "honorary academician" and the appointment of a pension of 200 rubles per year). On January 14 ( January 25 ), 1731, Herman left St. Petersburg and went to his native Basel ^[19] . In Basel, Herman continued to maintain a scientific relationship with the St. Petersburg Academy of Sciences and publish his works in its publications ^[20] .

In 1733, German was elected a member of the Paris Academy of Sciences , but died on July 14 of the same year ^[3] .

The main works of Herman relate to mechanics and analysis (with the application of the latter to geometry ), as well as to the history of mathematics. He developed the theory of integration of ordinary differential equations of the first order, the theory of curves and surfaces of the second order , studied the problems of integral calculus and elementary geometry , spherical epicycloids ^{[8] [21]} .

In mechanics, Herman studied the motion of bodies in a medium or in a vacuum under the action of variable forces , and studied the theory of gravity and external ballistics ^[22] .

Herman’s most outstanding work was ^[23] his treatise on dynamics “Foronomia, or on the forces and motions of solids and liquids” ^[24] , which he began to write in Padua and finished in Frankfurt an der Oder , publishing it in 1716 year (by "foronomy" German meant science, which later became known as " theoretical mechanics "). L. Euler praised Foronomia; in the preface to his first fundamental treatise, “Mechanics, or the science of motion, presented analytically” ( 1736 ), he put it on a par with Newton's works “Mathematical Principles of Natural Philosophy” and P. Varignon “New Mechanics, or Statics”. It was the three listed treatises that became the starting point for many Euler studies ^[25] .

The Hermann-Euler Principle

In chapter V of the second part of the book of the first “Foronomia,” Herman dealt with the problem of determining the reduced length of a composite physical pendulum (which is a set of several material points rigidly fastened together and capable of jointly rotating around a horizontal axis under the action of gravity ), developing in the process of solving it a special version of the principle of reducing the system’s motion conditions to the conditions of its equilibrium ^[26] (and anticipating the later d'Alembert principle ^[27] ).

An analysis of this problem (in the case of two point loads) was also carried out by Herman's teacher, Jacob Bernoulli. The similarity of the ideas of both scientists is evident from the similarity of the terminology they use: to denote the concept of “power” Herman uses the same term sollicitatio 'motivation' as J. Bernoulli ^[18] . Like the latter, Herman introduces for free points of a separate pendulum “free” and “true” motives for movement (that is, forces that cause free and true acceleration of these points, respectively). However, unlike his predecessor, Herman goes in reducing the dynamic problem to the static one in a different way, and does not base the theory of motion of a composite pendulum on the condition of the pendulum equilibrium under the influence of the “lost” motivations for motion (driving forces) applied to it, but the condition equivalence of two aggregates applied to the points of the pendulum of forces - true driving forces and free moving forces. Thus, the theory of motion of a composite pendulum in Herman's approach is significantly simplified (with the elimination of the need to form and use such additional scientific abstractions as the “lost” and “acquired” motivations for movement used by Jacob Bernoulli) ^[28] .

Instead, Herman introduces the concept of “vicar” (substitute) forces ( lat. Sollicitationes vicariae ) for gravity ^[29] ; as applied to the points of a composite pendulum, these are forces whose directions are perpendicular to the radius vectors of the points. Herman's replacement forces are, by definition, equivalent to the assigned forces (that is, the forces of gravity); This equivalence should be understood as follows: if the directions of all “substitute” forces are reversed, then the pendulum with the simultaneous action of the system of gravity and the new system of forces will remain in equilibrium ^{[27] [30]} .

Herman points out ^[31] : “For our case, consideration of the actual movement yields nothing, since in this case this movement, already acquired, should be considered as general, in which individual particles are carried away; but let us consider the increments of particle velocities instantly communicated to them, and this nascent movement can be investigated whether it is generated by “substitute forces” ... or by real gravity forces ” ^[32] .

Postulating this equivalence, Herman writes down the equivalence condition in the form of the equality of the total moment of true driving forces (vicar forces) relative to the axis of rotation of the pendulum to the total moment of free moving forces (gravity) relative to the same axis. Thus, in his main means of reducing a dynamic problem to a static one, it is precisely “replacement” forces that act, and not “lost” ones, as in J. Bernoulli; he does not calculate the latter and does not consider in detail (assuming the question of them has already been clarified), but only mentions ^{[28] [32]} .

Further, solving the problem, Herman proves two lemmas and proceeds to the proof of the main theorem, formulating it this way: if the point loads that make up the pendulum and moving under the action of gravity are mentally released from the bonds, then they will begin to move upward (each initially with that the speed that she received in the connected movement), and as a result, each of the cargoes can rise to such a height that the general center of gravity of the cargo system will again be at the height at which the associated movement began. It was from this position (accepted without proof) that H. Huygens proceeded when he built his theory of the physical pendulum ^{[29] [33]} .

In 1740, L. Euler in his memoir “On small vibrations of bodies, both solid and flexible. A new and easy method ” generalized Herman's approach (applied to only one specific problem) and used it to solve a number of different problems in the dynamics of systems of solids ^[29] . Euler briefly formulates the principle under consideration as the principle of equivalence of two systems of forces - forces “relevant” (that is, actually applied) and forces “required” (which would be sufficient to implement the same movement in the absence of connections), while clearly indicating the connection of the discussed approach and static methods. The Hermann – Euler principle formulated in this way was in fact a form of the D'Alembert principle — moreover, found earlier than the publication of D'Alembert's “Dynamics” ( 1743 ). However (unlike the D'Alembert principle), the Herman – Euler principle has not yet been considered by its authors as the basis of a general method for solving problems on the motion of mechanical systems with constraints ^{[34] [35]} .

Note that in the Petersburg period of his life, German again returned to the problem of the physical pendulum and solved it (in another way) in the article “A new way to derive the rule for determining the center of vibration of any complex pendulum, already considered, obtained from the theory of the motion of heavy bodies along circular arcs” (presented to the Academy of Sciences in 1728) ^[36] . His conclusion, in essence, coincides with the usual proof of the mentioned rule using the integral of living forces ^[29] .

In 1935, the International Astronomical Union named Herman as a crater on the visible side of the moon .

• Hermann J. Responsio ad cl. Nieuwenteyt considerationes secundes circa calculi differentialis principia. - Basel, 1700.
• Hermann J. Methodus inveniendi radios osculi in curvis ex focis descriptis // Acta Eruditorum . - Leipzig: Grosse & Gleditsch, 1702. - P. 501-504.
• Hermann J. Demonstratio geminae formulae and Celeberrimo Dn. Joh. Bernoulli pro multisectione anguli vel arcus circularis, sine demonstratione exhibita // Acta Eruditorum . - Leipzig: Grosse & Gleditsch, 1703. - P. 345-360.
• Hermann J. Denova accelerationis lege, qua gravia versus terram feruntur, suppositis Motu diurno terrae et vi gravitatis constanti // Acta Eruditorum . - Leipzig: Grosse & Gleditsch, 1709. - P. 404-411.
• Ermanno J. Metodo d'investigare l'Orbite de 'Pianeti, nell' ipotesi che le forze centrali o pure le gravit'a degli stessi Pianeti sono in ragione reciproca de 'quadrati delle distanze, che i medesimi tengono dal Centro, a cui si dirigono le forze stesse // Giornale de 'letterati d'Italia , 2 , 1710. - P. 447-467.
• Hermann J. Phoronomia sive de viribus et motibus corporum solidorum et fluidorum. Liberi duo . - Amstelædami: R. & G. Wetstenios, 1716.
• Hermann J. De mensura virium corporum // Commentarii Academiae Imperialis Scientiarum Petropolitanae . Tomus I. - St. Petersburg. , 1728. - P. 1–42.
• Hermann J. De calculo integrali // Commentarii Academiae Imperialis Scientiarum Petropolitanae . Tomus I. - St. Petersburg. , 1728. - P. 149-167.
• Hermann J. Nova ratio deducendi regulam jam passim traditam pro centro oscillationis penduli cujusque compositi petita ex theoria motus gravium in arcubus circularibus // Commentarii Academiae Imperialis Scientiarum Petropolitanae . Tomus III. - SPb. , 1732. - P. 1-12.

• Herman, Jacob // Brockhaus and Efron Encyclopedic Dictionary : in 86 volumes (82 volumes and 4 additional). - SPb. , 1890-1907.
• Bogolyubov A.N. Mathematics. The mechanics. Biographical reference. - Kiev: Naukova Dumka , 1983 .-- 639 p.
• Veselovsky I.N. Essays on the history of theoretical mechanics. - M .: Higher school , 1974. - 287 p.
• History of Mechanics in Russia / Ed. A.N. Bogolyubova , I.Z. Shtokalo . - Kiev: Naukova Dumka , 1987 .-- 392 p.
• Moiseev N. D. Essays on the history of the development of mechanics. - M .: Publishing house Mosk. University, 1961 .-- 478 p.
• Pekarsky P.P. History of the Imperial Academy of Sciences in St. Petersburg. T. 1 . - SPb. , 1870. - LXVIII + 774 s. Archived July 3, 2014. Archived July 3, 2014 on Wayback Machine
• Tyulina I.A. History and methodology of mechanics. - M .: Publishing house Mosk. University, 1979. - 282 p.

• Profile German Jacob (Jacob) on the official website of the RAS
• O'Connor JJ, Robertson EF Jakob Hermann (Neopr.) . - Materials archive MacTutor . Date of treatment August 5, 2013.