A formula is developed connecting the wingbeat frequency of insects with their masses and wing areas. It is derived first theoretically, using dimensional analysis, and then it is compared with published measurements. The formula discovered involves two parameters which dimensional analysis alone cannot determine. One of these is evaluated using one among many proposed semi-empirical relationships (the only one that stands scrutiny); the other by fitting a published dataset. It is found that the resulting equation, applying to insects in general, accords well with observation, and indeed is very close to being optimal (in a sense to be defined).

This paper, the first of two, follows the development of the LAPLACE Transform from its earliest beginnings with EULER, usually dated at 1737, to the year 1 8 8 0 , when SPITZER was its major, if himself relatively minor, protagonist. The coverage aims at completeness, a n d shows the state which the technique reached in the hands of its greatest exponent to that time, PETZVAL. A sequel will trace the development of the m o d e r n theory from its beginnings with POINCARI~ to its present form, due to DOETSCH.Theorie und Anwendung der Laplace-Transformation [18] in 1937. Its ready acceptance into undergraduate mathematics courses is a remarkable story, to be taken up elsewhere.However, the theory has a long history, dating back to EULER, whose first paper on the subject [20] appeared in 1737. This paper and its sequel concern themselves with the two hundred years between this paper and DOETSCH' book.The year 1880 is a natural place at which to interrupt the account as it is here that POINCARt~ introduces to the theory the full power of the calculus of residues.Previous histories of the LAPLACE Transform have been published. The earliest is by SPITZER in the introduction to his Vorlesungen iiber lineare Differential-Gleichungen (1878) [89]. This is relatively complete for its time, but is written tendentiously, and with a view to establishing LAPLACE'S priority. It neglects (one is tempted -see Section 3 below -to say "deliberately neglects") the prior work of EULER. PINCHERLE gives two accounts [71,72], both predating DOETSCH, and neither of them at all complete. A recent paper by PETROVA [64] discusses the work of LAPLACE and of ABEL, but is incorrect in many of its details. The best account extant is that of GRATTAN-GUINNESS [31 ], although this, largely on account of its brevity, is also incomplete. DOETSCH' Kapitel 2 and Historische Anmerkungen [18, pp. 6-12, 404-424] and the associated bibliography provide much useful material, particularly for the second of the two periods , and there is also valuable material to be found in works by SCHLESINGER [78,79].Of the many potted histories to be found in modern texts, mention need only be made of the best -that of CARSLAW • JAEGER [7]. It is typical of most such accounts in that it concentrates almost exclusively on a line of development beginning with HEAVISIDE, and ending with VAN DER POE, and thus neglects entirely the mainstream of the theory's development. As a result of this eccentric approach, it sees the development of the LAPLACE Transform as an attempt to rigorize the Calculus of Operators, although, in fact, the two branches of theory are quite distinct for much of their history. This paper and its sequel attempt to provide a full account of the development of the LAPLACE Transform, not only to complete and correct earlier histories, but also to provide, in convenient form, a coherent narrative of both mathematical and human interest, Attention will focus primarily on the use of the LAPLACE Transform in the solution of linear differential equations, as t...

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Considerations of dimensional analysis are used to derive a formula connecting wing-beat frequency with mass and wing area for a flying insect. The formula is used to demonstrate the existence of asynchronous flight muscle.