**HERMANN'S **

Ian Bruce

This is a work which has never
been translated into English, apart from a few sections ; it forms a bridge
between Newton's Principia and Euler's Mechanics; indeed Euler and Hermann,
distantly related, worked together at St. Petersburg for a few years, but long
after the present work was composed, around 1712 and published in 1715. Some of
Hermann's biographical details can be found in Wikipedia, so we will not linger
over these here, except to say he was a student of James Bernoulli, was held in
high regard by Leibniz, he had an understanding of physics from his work in
Italy, and eventually worked at St. Petersburg, alongside Euler ; it is a
difficult work to translate for a number of reasons; the more salient being the
poor state of the diagrams, most of which are incomprehensible without a lot of
work, there are numerous typographical errors in the text and diagrams : I
mention here with gratitude the assistance provided to me by the library of the
University of Ghent, where a helpful individual provided me with a complete set
of all the figures in the *Phoronomia*,
all 160 of which I have redrawn]. Hermann's method of reasoning lies somewhere
between his two idols, Newton and Leibniz : this, of course, is what renders
his work interesting ; it lets us know where matters stood, at least on the
continent of Europe, at this time, at least for this individual. The first 3 chapters
deal with statics in the first section of Book I, while the second section is
devoted to dynamics, which comes as rather a shock, as his methods seem to be
in advance of what was regarded as true at the time; in fact some of his
equations can be considered as nothing less than the conservation of potential
and kinetic energy: he arrived at these via Newton's second law set out with
the acceleration *a = vdv/dx*, which enables one to bypass the use of time in
equations involving acceleration. Ch. VI of Book I, Section II is interesting
in this regard; some of the ideas have come originally from Huygens.

Book II is concerned with Hydrodynamics, and historically it was the first work to tackle this science using Leibnizian Calculus, at least in part. Some of the basic ideas are wrong ; for example, the pressure in a static liquid does not depend on the total weight of the supported fluid, only on the depth; however, in a translation such as this, the idea is to get the thoughts of the author across, not to correct his work , unless of course it is a simple error. Section I is concerned with hydrostatics, while Section II is concerned with hydrodynamics, initially without friction of any kind. Some interesting results emerge in this section, essentially the existence of a form of the Bernoulli Principle for fluid flow, originating in the works of Castelli, Baliani, Torricelli, Borelli, and used by Gulielmini to estimate the flow of water in rivers and channels. Book III considers the effects of resistance to the passage of a body, both parallel to the axis and at an angle to the axis. Section IV considers the effect of air resistance on bodies moving through air, under increasingly complicated situations, and where the resistance can vary in different ways; most of the working uses arcane geometric methods similar to those presented by Newton in his Principia. The appendices deal with some corrections and matters not handled in the text, as you can find out below.

References :

The work itself
can be found in the *Garden of Archimedes*
series on cd no. 1.

A complete listing
of Hermann's works can be found in *A Catalog of the Works of Jacob Hermann*
(1678-1733), by Fritz Nagel, in *HISTORIA MATHEMATICA* 18 (1991), pp. 36-54.

The Latin text by
Baliani mentioned in Book 1, Sect. 2, Ch. 1, in which the equations of
accelerated motion are first considered, can be found at the e-rara website.

*Reading the Principia*, by Niccolo Guicciardini, contains reference to Hermann's *Phoronomia*, and provides another point
of view regarding Prop. I of the Principia, where Newton established his area
law, on p. 211 onwards, which Hermann derives independently in Ch. 2 of Section
2, Book I.

*Jacob Hermann and the Diffusion of the Libnizian Calculus in
Italy* by Mazonne and
Roero is a wonderful work in my opinion, as it contains a great deal of
original material on the early days of the calculus; there is much mention of
the *Phoronomia*, but no intimate
details, as the work had never been translated until now as a whole. It was
regarded as being hard to understand.

BOOK
ONE: SECTION 1

Click here for the Preface
and some introductory notes (§1−§25).

Click here : Ch's. I &2. Chapter
1 (§26−§34) is about levers and moments of forces, and shares a lot with
Varignon's Mechanics, which had appeared a few years earlier. The conditions
for equilibrium of a body under various forces is considered. Chapter 2 (§35−§83)
considers forces of a continuous nature acting on a body of a generalized shape
; at present the description is general, but the body is regarded as quite
rigid, and might represent the hull of a ship in the sea, the surface of a sail
blowing in the wind, etc.

Click here : Chapter 3 (§84−§113). This is an
interesting chapter, as the curves formed by a flexible string or wire under
various loading situations are investigated ; this is done geometrically at
first, and finally Leibniz's calculus is used to find the equations in analytic
form. The curves formed from extended two dimensional sheets such as sails are
also investigated.

BOOK ONE : SECTION 2.

Click here : for Chapter 1 (§114−§152). This is a
most interesting chapter. Hermann clearly had an excellent understanding both
of Newton's work, as well as Leibniz's calculus. Here he sets the foundations
for future chapters on dynamics, were he demonstrates in a rather antiquated
notation a number of results which have survived the test of time, and amount
to the conservation of energy, although of course his equations do not make
this connection; nevertheless the equations are there. He even gets ½ mv^{2}
for the work done by gravity for a falling body: thus future workers would have
done well to have considered his work in more detail; his approach is
essentially modern, yet who has even heard of him today?

Click here : for Chapter 2 (§153−§169). This is
another most interesting chapter, packed with Hermann's way of finding the
curve traced out by a mobile body obeying a general law of gravitation. As
usual, a lot of information is packed into the diagrams, esp. Fig. 37 : which
shows together diagrams for the force, velocity, displacement, changes in the
tangents, etc. for a body moving relative to a fixed attracting body.
Eventually, a general law is produced for the orbit of such a body; later, he
specializes with algebraic curves of various kinds as examples. I have not
worked through the work in the *Acta Erud.*
referred to, regarding the dissection of angle, so this is taken on trust at
present; if this is of particular concern to anyone, I can probably translate
these pages : finding diagrams is always a problem, however. Google is a great
disappointment in this regard : why go to the trouble of scanning a book if you
miss out fold-out diagrams? As somebody said, they are more interested in
quantity than quality: for it negates the whole effort – rather like selling
you a car without the wheels, which can't be had for love nor money……

Click here : for Chapter 3 (§171−§186). I have
become quite attached to this chapter, and I am sorry to have to leave it; in
it you will find much information about gravitational isotones in general,
before Hermann delves into the cycloid and epicycloid by his own mainly
geometrical methods, which he relates finally to calculus : essentially he had
found the work energy relations involving potential and kinetic energy;
however, because these physical concepts were unknown at the time, their mathematical representations remain
just that, and no attempt was made to put things onto a physical footing. Do not
however, expect an 'easy read', though I have tried to lighten the load via
notes.

Click here : for Chapter 4 (§187−§196). This
chapter is taken from the *Principia* mainly
as Hermann indicates, about the motion of apses, with a diversion into
Varignon, who handled the question of apses from the Cartesian/Huygens
viewpoint, though that point is not made clear, and unfortunately centrifugal
forces are introduced. Interesting from the early calculus point of view.

Click here : for Chapter 5 (§197−§211). This
chapter is only of academic interest, I think ; a compound pendulum is immersed
in a fluid, the parts of the pendulum of equal volume are made from materials
with different densities to each other and to the fluid, and the equivalent
simple pendulum is found; no account is taken of fluid resistance, etc. Hermann
shows that in these circumstances the centre of oscillation and percussion of
the pendulum are different, which of course is noteworthy.

Click here : for Chapter 6 (§212−§237). This
chapter is full of interesting material, (but not all of which is immediately
accessible to the modern reader, as almost everything is handled in terms of
ratios), for example : *Whatever the
masses, in an elastic collision in one dimension between a moving body and one
at rest in the lab. ref. frame, the latter moves off with twice the speed of
the C. of M., which remains at a constant speed throughout*. Hermann had
been looking at one of Huygens' posthumous works, *de Motu Corporum ex Percussione*, [The motions of colliding bodies]
and presented solutions to the propositions, which are presented without proof.

BOOK TWO : SECTION 1.

Click here : for Chapter 1 (§238−§262). This is
the introductory chapter, where definitions etc. are set up, and some theorems
are developed relating pressure to depth of fluid, density, etc.

Click here : for Chapter 2 (§263−§289). Here
Hermann sets out his thoughts on the pressure exerted on the walls of a
chamber, which appear to be surprisingly accurate in parts ; however, he seems
to have been led astray in his analysis of deformable chambers; not all, I must
confess, I have understood completely according to his line of thinking,
especially his last geometric proposition; and of course one cannot take
moments in liquids as if they were solids under stress, etc.

Click here : for Chapter 3 (§290−§302). This
chapter is concerned with the equilibrium of immersed or bodies floating in a
fluid. Unfortunately, there is no mention of metacentric height, so that the
presentation is rather limited, and occasionally quite wrong.

Click here : for Chapter 4 (§303−§311). This
chapter is concerned with the deformation of flexible vessels, such as the
shape adopted by a sail filled with water, which is found to agree with the
calculations of the Bernoulli brothers.

Click here : for Chapter 5 (§312−§324). This chapter
is concerned with barometers mainly, and is largely descriptive. Refinements
are made to barometers to make them more sensitive to atmospheric changes.

Click here : for Chapter 6 (§325−§338). This
chapter is concerned with the construction of vacuum pumps mainly, with some
early views on the nature of pressure. I have added some pages from Boyle's
work describing the construction of his pump. Quite interesting.

Click here : for Chapter 7 (§339−§353). This
chapter is concerned with showing the proportionality between pressure and
density of air; it follows Newton's approach as cited in the text.

Click here : for Chapter 8 (§354−§383). This
chapter is concerned with showing the proportionality between pressure and
density of air, in which different models are examined. This is a continuation
of the last chapter, and has some interesting applications to measuring heights
of mountains, used by Cassini in his cartography of France.

BOOK TWO : SECTION 2.

Click here : for Chapter 9 (§384−§406). This
chapter is concerned with the flow of water from vessels, and a model is
developed which is a fore-runner of what is now called the Bernoulli Principle;
however, although formulas are obtained by which the flow from a simple hole or
from a larger section in the wall of a vessel maintained at a fixed level with
water can be calculated, these do not involve the conservation of energy as
such, which lay in the future, but which amount to the same thing. The
arguments depend on the final velocity of the water out flowing being
calculated from the time for a weight to fall without friction through the same
height; clearly the water does not perform this motion, but does so more
slowly, yet conserving energy, if no frictional forces are present. The
filaments introduced would seem to be analogous to streamlines. I have not gone
to the trouble of transposing the arguments into the terms of modern calculus,
which means it may be hard to follow at times, but it seems to be better
to leave matters as presented.

Click here : for Chapter 10 (§407−§420). This
chapter is a continuation of the previous one, but establishes the use of the
final theorems therein; it is an attempt to provide solutions to real problems
involving hydrodynamics, such as gauging the amount of water flowing though a
sluice gate into a channel, etc., as well as investigating the speed of water
in an inclined channel, etc.

BOOK TWO : SECTION 3.

Click here : for Chapter 11 (§421−§433). Hermann is
now ready to consider forces exerted between fluids in contact other than
pressures induced by their weight, and the interactions between fluids and
solid bodies, whichever is moving. He presents some theorems that may seem
naïve, and one is left wondering whether he considers the collisions to be
elastic or inelastic and no explanation is given of what the fluid does after
the collision : he considers it merely to 'slip away' ; no attempt is made to
consider the viscosity or resistance at this stage; however, he produces a
formula for what he considers the most efficient angle to extract motion from a
moving fluid, such as windmill sail in the wind, which is interesting.

Click here : for Chapter 12*a* (§434−§462 for whole chapter). I have split the present
chapter into two parts, as the going is rather heavy, and a lot of new material
is presented. The work deals with the ratio of the resistances experienced by a
body moving along its axis where the shape is some given curve, or a curve to
be found, where the vertex leads or the base of the curve leads. The resistance
is taken proportional to the square of the speed. A lot of the material can be found in *Elements of Hydrostatics* published by
Miles Bland, a Cambridge lecturer, in 1827, and available from Google *with its diagrams* : without any
reference to Hermann at all! Here the advances in analysis and algebra are
evident, and a good historical comparison can be made. The reader may wish to
consult this book to aid understanding, where I have not provided notes.

Click here : for Chapter 12*b*. Hermann now considers uniform motion of a fluid at an angle to
the main axis of a body, or conversely the resistance of the same body not
moving along its main axis, but at some angle, through a fluid at rest. The
analysis considers the impressed force to be separated into horizontal and
vertical components at the incremental level, then summed or integrated to find
the equivalent vertical and horizontal forces, acting on an arbitrary convex
curve. A number of examples are then detailed for well-known curves, in which
the necessary calculations are performed. The texts I have used have poor
printing, so that there may remain the occasional misprint, letters are
occasionally lacking in the diagrams, which I have inserted.

Click here : for Chapter 13 (§463−§476). Hermann
now considers the shape of a sail in the wind; he relies on the analysis of
Johan Bernoulli to produce his own version based on the catenary, but he
elaborates greatly on the derivation of the results both geometrically as well
as analytically. There is a basic problem trying to put differential elements
on a diagram, as squares are no longer squares, a point on a tangent line may
lie inside the circle, etc. Use is made of the equilateral hyperbola rather
than the equiangular hyperbola in working out the areas of sectors of the
hyperbola in terms of a logarithmic curve, the inverse of which was the
catenary curve. I have provided notes for the first proposition mainly; there
is a lot of interesting mathematics in this chapter, not all of which I have
time to investigate at present ……

BOOK TWO : SECTION 4.

Click here : for Chapter 14 (§477−§494) . Hermann
now sets out some properties of his
logarithmic curve that he is going to find useful in the following chapters, in
which he intends to find the motion of bodies in mediums with the resistance
proportional to the speed, the square of the speed, etc. I have indicated that this will probably be
incorrect, as such motions depend on the exponential function, not yet
developed by Euler ; however, it will be of some interest to see what
transpires; Hermann's logarithmic curve seems to be elusive at present, as it
cannot be the graph of simple logarithms, as he seems to be indicating. Perhaps
some knowing person can enlighten me ……

Click
here : For Huygens thoughts on the logarithmic
curve. Well, nobody enlightened me but I have worked it out myself, or rather
Huygens did, in a manner of speaking, in a note entered at the end of his *Treatise on Optics*, which you can look
at here. The curve in question is the exponential curve, to some base *a *in general, or *e, *in which case the subtangent has unit length always.

Click here : for Chapter 15 (§495−§521) . Hermann
now sets out to establish the formulas for the acceleration, velocity, and
distance gone, for a body projected downwards, upwards, at an angle, and at an
angle to an inclined plane where the air resistance is proportional to the
speed ; in which he appears to be successful in establishing the requisite
formulas. He also demonstrates that his
method produces the same results as Huygens, Varigon, and Newton obtained by
different methods, so there is little doubt that he is correct. However,
explanations are lacking, the same exponential decay curve is used for each
kind of motion, and there is some confusion about what the various lines
represent. Part of the trouble is the lack of analytical formulas to supplement
the mainly geometric style of proof. It is now clearer to me that Hermann spend
considerable ingenuity is establishing geometrical arguments and finally taking
what he called moments, to effect the differential equations and their
integrations which we now assert at once from conservation of mechanical
energy, work done by friction, etc. In a sense, he was 100 years ahead of his
time, as these principles were unknown at the time. The natural logarithm curve
forms the backbone of these calculations, and that equates an arithmetic
progression of the times to the log of a geometric progression of the speeds,
while the inverse logarithmic curve [i.e. exponential] turns a geometric
progression of the speeds into the anti-log of an arithmetic progression of the
time, as Huygens had surmised. I am at present making small adjustments to
these latter chapters by way of notes, etc.

Click here : for Chapter 16 (§522−§539) . Hermann
now sets out to establish the formulas for the acceleration, velocity, and
displacement for situations similar to the above, but in which the air
resistance varies as the square of the speed.

Click here : for Chapter 17 (§540−§561) . Hermann
now sets out to establish the formulas for the acceleration, velocity, and displacement
for situations similar to the above, but in which the air resistance varies as
the square of the speed but also includes a term proportional to the speed to
account for the viscosity of the air. The propositions are presented
geometrically, and are hard to understand at first; it is probably a good idea
to have a look at the next chapter while reading this chapter, as the
motivation for the geometrical procedures adopted can be understood in terms of
presenting logarithmic integrals that can be evaluated. There is a problem with
the viscous term, as the two complementary physical situations presented in the
text do not have the same resistive forces acting ; we know for example, that
if a ball is thrown upwards, it takes longer to come down than to go up; I am
not sure if Hermann appreciated this fact in his discussions.

Click here : for Chapter 18 (§562−§580) . Hermann
now sets out to establish the formulas for the acceleration, velocity, and
displacement for situations similar to the above analytically; this is a great
help in understanding the propositions presented above. One wonders why he went
to the bother of presenting geometrical proofs, which in the end have to be
performed analytically in any case. The case where the density of the air is
varied is also considered here.

Click here : for Chapter 19 (§581−§601) . Hermann
now applies his theory do motions on special curves, and he concentrates on the
cycloid, with the air resistance proportional to the square of the speed. I
have indicated in a note the pre-energy vs work done ideas used by Hermann in
his analysis, which were not appreciated at the time.

Click here : for Chapter 20 (§602−§623) . A body
is projected along a given curve with air resistance proportional to the square
of the speed, while it is attracted by masses located at some points on another
curve. The forces at any point on the former curve are resolved and integrated
to describe the motion. However, the analysis is performed geometrically,
and a number of limiting cases are
examined, and in which generally the rectangular hyperbola is used to change ratios
of quantities into areas and vice versa, essentially the logarithms of the
ratios, although logs are not referred to in the text. As in a lot of Hermann's
work, no particular problem is actually solved numerically: instead, curves are
sketched generally and the analysis is geometrical. In other words, what you
see is what you get, as geometry is essentially a visual form of mathematics,
devoid of actual numbers.

Click here : for Chapter 21 (§624−§629). Here the
air resistance of the wind on a sail, taken proportional to the square of the
relative speed of the wind to the sail, and with the water resistance removed
in a similar manner, is taken as the
accelerating force generating essentially the kinetic energy of the
ship.

Click here : for Chapter 22-24 (§624−§629). In Ch.
22 a number of results are established for the shape of the earth, similar to
those developed by Newton and Huygens. Ch. 23 enlarges on Newton’s method for
finding the speed of sound, and employs some of the geometry associated with
simple harmonic motion. Ch. 24 gives the first indication of the kinetic
theory, with the pressure depending on the mean speed squared of the particles
and the density of the gas.

Click here : for the Appendix to Book I. A number of
issues are investigated here in more detail, especially Kepler's Laws from an
analytical point of view, and the calculation of the position of the centre of
oscillation of an extended body, a forerunner of the moment of inertia analysis
of Euler. Unfortunately, I have not the time to spend doing a thorough analysis
of these matters.

Click here : for the rest of the Appendix. A number of
items are added here finally. An analysis of Hermann's interpolation schemes;
some notes on the rate of flow of a liquid from the bottom of a vessel, erroneous
of course, and the correction of Prop.
XXV, Book I, where an incorrect sign had crept into the calculation.

Ian
Bruce. 16^{th} June, 2016; latest revision.
Copyright : I reserve the right to publish this translated work in book form.
You are not given permission to sell all or any part of this translation as an
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interest, you can copy part or all of the work for legitimate personal or
educational uses. See note on the index page.