The First Part Containing Book One,


Concerning the Explanation of the Principles, from which the Construction both of Telescopes

as well as Microscopes is Desired.

     This is one of Euler's major works, extending over most of his working life; the chapters in the three volumes were produced mainly from his shorter articles produced over many years, and assembled with the help of his co-workers. At present,  volumes one is now complete: the first chapter is concerned with the spreading of the image by a single thick convex lens, (essentially the first ever treatment of spherical aberration), while the second considers the spreading of the image by a number of such lenses on the same axis, with attempts to minimize this effect for two thin lenses .  Ch. 3 is rather long, but gives a thorough discussion of how to minimize the spreading or confusion of the final image for two, three or four thin lenses. The case for 4 lenses turns out to be especially relevant. Ch. 4 is a compilation of the preceding chapters, and considers the combined effects on the image viewed by the eye due to magnification, confusion of image, and clarity. Ch. 5 is concerned with determining the field of view of an object seen through a number of lenses, and a convenient place for the eye. Unfortunately, the treatment of the eye's accommodation was not understood at this time, and Euler considers the eye as a camera obscura. In addition, the treatment of image formation is rather obscure in this work up to this point.  Ch. 6 is long and rather involved; in it Euler sets out formulas for the distance and height of the image formed by a series of lenses. Note that here and in the previous chapters, some of the lines do not refer to rays, but to measurable lengths. The effect of lenses of differing refractive indices is introduced, and finally a method is produced for producing an achromatic final image from a series of such lenses, as well as being free of confusion; these may be viewed by the modern reader with some suspicion. Ch. 7 is a summary of the preceding chapters, to which I have added the occasional notes.

chapter I  : Concerning the spreading of the image shown by a single lens.

chapter 2  : The spreading of the image presented by several lenses.

            chapter 3  : Concerning multiple or composite lenses.

            chapter 4  : Concerning the confusion of vision both by the apparent magnitude and clarity of the image.

            chapter 5  : Concerning the apparent field of view, with the eye at the most suitable place.

            chapter 6  : Concerning the confusion arising from the nature of diverse rays.

            chapter 7  : Concerning the construction of dioptric instruments in general.

             It is in this chapter in particular that Euler advocated the removal of the colored fringes round the image of an object viewed through a combination of lenses, by careful adjustments of the stops or diaphragms used : this assertion detracted considerably from the value of the work, as it was of course complete nonsense.

E266 achromatic doublet  : Construction of objective lenses from two glasses…..

The Second Part Containing Book Two :


Section I

Ch.I  is an updated summary of Book I. Whether he understood Dollond's new approach for making achromatic lens doublets is not clear; though some clarification has been made of other matters. Ch. 2 more or less gives Euler's recipies for making perfect composite objective lenses for telescopes; i.e. those with the minimum confusion or spherical aberration and free from colored fringes or achromatic; two or three lenses of differing refractive indices are used. Ch. 3 gives an attempt to classify types of telescopes according to the number of real intermediate images produced. Ch. 4 is concerned with perhaps the simplest of telescopes, namely a convex objective followed by a concave eyepiece ; this kind of telescope is treated at length, although it would appear to be at variance with the actual devices produced by Dollond; yet there appears to be much of value in this text, though Euler seldom gets a mention in optics text re telescopes. Ch. 5 is now complete, which shows how to perfect simple upright image telescopes by the use of extra lenses; this is rather a long chapter.

            chapter 1  : Telescopes in general.

            chapter 2  : Telescopes with composite perfect objective lenses.

            chapter 3  : The division of telescopes into three particular kinds.

            chapter 4  : Telescopes of the first kind are without real images and represent objects upright.

            chapter 5  : Concerning the further perfection of telescopes of the first kind, by the addition of one

                               or more lenses  .


    Book II, Sect. 2, Ch. 2  considers the advantages of inserting an extra lens to coincide with the primary image formed by the objective lens, and the nature of this lens to reduce confusion (i.e. aberrations); glass of the same refractive index is considered throughout at this stage. Ch. 3 considers further improvements using two kinds of glass with differing refracting indices, forms of crystal and crown glass. A formula for the dispersion of each kind is introduced and applied to the individual lenses; as mentioned above, defects in the final design are attributed to the poor artificier employed to implement the designs, which did not appear to work very well, and did not produce the sharp colour free images intended.

Sect. II, chapter 1  : The construction of telescopes of the second kind, with a convex objective and the inverse image .

Sect. II, chapter 2  : Concerning the further perfection of telescopes following the use of a single kind of glass.

Sect. II, chapter 3  : Concerning the further perfection of telescopes following the use of a two kinds of glass.


The translation of Part 3 of Book 2 is now complete, and we can now present Ch.1, Ch.2 and Ch.3, the former is concerned with producing upright images, by the addition of extra lenses. I have included some notes here. Ch.2 is occupied with terrestrial telescopes:  Euler's work was continually frustrated by the imperfect telescopes his artisans could make from his theoretical calculations; the flaw would appear to arise from his use of a formula for the focal length of the objective which he produced in Ch. I of the first volume, long before the different forms of lens aberration were understood. As it was, up to this point, he had not made use of Dollond's achromatic doublet, but used a separated pair of lenses, which did not produce very good results. In Ch.3, Euler is concerned with adapting his theory of terrestial telescopes to the innovations of John Dollond, where the chromatic aberration is reduced by using two lenses of differing refractive index, the first concave and the latter convex. Finally, some worked examples are set out to ease the burden of understanding the theory, and to make the work more understandable. The appendices on reflecting telescopes remains to be translated.

Sect. III, chapter 1  : The construction of telescopes of the third kind, for which objects again are represented erect.

Sect. III, chapter 2  : Concerning common terrestrial telescopes and their perfection.

Sect. III, chapter 3  : Concerning other kinds of telescopes of the third kind and their perfection.


The translation of the appendices of Book 2 is now complete, detailing the different kinds of telescopes involving spherical mirrors.

Appendix I  : Concerning common terrestrial telescopes and their perfection.

Appendix II  : Concerned with dual telescopes : objective mirror with a central hole plus smaller concave/convex mirror, and with focusing lenses for the eyepiece; discussion of the associated confusion (spherical aberration on axis) and their perfection.

Appendix III  : Concerned further with an example of a catadioptric telescope : initially some notes are provided on Euler's algebraic method of analysing such optical systems.

Appendix IV  : Concerned with  catadioptric telescopes constructed with a convex smaller mirror.

The Third Part Containing Book Three :


Section I

Chapter 1  :

Ch.I  is an purely introductory; however, it does contain a review of Euler's analytical method of treating optical problems.

Chapter Ia  :

Ch.Ia  is is the actual first chapter; this contains much interesting material on single lens microscopes.

Chapter 2  :

Ch.2  gives a detailed introduction to the construction of simple microscopes involving a number of lenses,

Chapter 3  :  provides the formulas for the lenses to be used, according to Euler's theories, either  arising from the aperture size, and the use of one or two refractive indices for removing the confusion arising from the approx. image position.

Section II

Chapter 1  : This chapter does not have the heading Ch.1 as it is the only one present. An  concave lens is either present as the eyepiece or is added to the objective, and has the title: Microscopes In Which No Real Images Occur. Various systems are considered, including  cases in which crown and crystal glass is used for a pair of lenses in the objective; Euler does not consider the achromatic doublet, which had been developed at this time, and he still erroneously considered removing the coloured margins from images, rather than removing the confusion thoroughout the image by such a lens.

Section III

Chapter 1  : This chapter is concerned with simpler composite microscopes, in which a single real image is to be found, to which all microscopes of this kind are referred.

Chapter 2  : This chapter is a long chapter concerned with more complicated composite microscopes, in which a number of lenses are used to form the objective, resulting in higher possible magnifications. A number of such microscopes, and also telescopes are set out from the principles established here.

Chapter 3  : This chapter is an addition to the second chapter, in which a concave lens is added to the objective, which is now a compound lens, in order to reduce the confusion of the image and to remove the colored fringe, at least according to Euler.

Chapter 4  : This chapter is a further addition in which the field of view is extended further, in addition to the benefits outlined above.

Section IV

Chapter 1  : This chapter is concerned with composite microscopes with two intermediate real images ; an extension to terrestrial telescopes is made here.

Chapter 2  : This chapter is concerned with further composite microscopes with two intermediate real images ; again, an extension to telescopes is made .

Chapter 3  : This final chapter is concerned with the further perfection of compound microscopes and with the confusion removed.

Ian Bruce. June 8th, 2020 latest revision. Copyright : I reserve the right to publish this translated work in book form. However, if you are a student, teacher, or just someone with an interest, you can copy part or all of the work for legitimate personal or educational uses. Please feel free to contact me if you wish, especially if you have any relevant comments or concerns about this work. My email address can be constructed from the information at the bottom of the introductory page of this website.