[In the following excerpt, Crombie demonstrates the influence of Robert Grosseteste 's thought upon Bacon 's scientifictheories.]

The writer who most thoroughly grasped, and who mostelaborately developed Grosseteste's attitude to nature andtheory of science was Roger Bacon (c. 1214-92) himself. Recent research has shown that in many of the aspects ofhis science in which he has been thought to have beenmost original, Bacon was simply taking over the Oxfordand Grossetestian tradition, though he was able also tomake use of new sources unknown to Grosseteste, as, forexample, the Optics of Alhazen. Though it is improbablethat Bacon heard Grosseteste's lectures at Oxford, heseems to have become a member of Grosseteste's 'circle'by 1249, and when he became a Franciscan friar he would, no doubt, have had access to his manuscripts. Baur hasdrawn attention to the many striking parallels between thescience of Roger Bacon and that of Grosseteste. The chiefpoint of resemblance to be noted is Bacon's 'grundsatzliche methodische Auffassung der Naturwissenschaft, unddie Erklarung des Wirkens und Werdens in der Natur'. Ithas been suggested above that Grosseteste in his search for reality and truth began with the theory of science which he developed in his commentary on Aristotle's Posterior Analytics, and that he then made use of this theory in his detailed scientific studies. Roger Bacon in his major writings on natural science, as in the Opus Maius, Opus Minus and Opus Tertium, the De Multiplicatione Specierum, and the Communis Mathematica and Communium Naturalium, also first postulated a theory of science as a means of discovering reality and truth, and then used this methodological theory in detailed researches undertaken as an illustration of it as well as for their own sakes. In setting out this theory of science Roger Bacon, like Grosseteste, began with Aristotle's Posterior Analyticsand he developed particularly those points to which Grosseteste had paid attention: the means of arriving at universals or causes by induction and experiment, and the use of mathematics as the most certain means of demonstrating the connexions between events.

Having laid down the fundamental principles ofthe wisdom of the Latins so far as they are found in language, mathematics and optics, [he said in Part VI of the Opus Maius, 'De Scientia Experimentali'] I now wish to unfold the principles of experimental science, since without experience nothing can be sufficiently known. For there are two modes of acquiring knowledge, namely, by reasoning and by experience. Reasoning draws a conclusion and makes us grant the conclusion, but does not make the conclusion certain, nor does it remove doubt so that the mind may rest on the intuition of truth, unless the mind discovers it by the method of experience (via experientiae); for many have the arguments relating to what can be known, but because they lack experience they neglect the arguments, and neither avoid what is harmful, nor follow what is good. For if a man who has never seen a fire should prove by adequate reasoning that fire burns and injures things and destroys them, his mind would not be satisfied thereby, nor would he avoid fire, until he placed his hand or some combustible substance in the fire, so that he might prove by experience that which reasoning taught. But when he has actual experience of combustion his mind is made certain and rests in the full light of truth. Therefore, reasoning does not suffice, but experience does.… What Aristotle says therefore to the effect that the demonstration is a syllogism that makes us know, is to be understood if the experience of it accompanies the demonstration, and is not to be understood of the bare demonstration.

This experimental science [he went on] has three great prerogatives with respect to the other sciences. The first is that it investigates by experiment the noble conclusions of all the sciences. For the other sciences know how to discover their principles by experiments, but their conclusions are reached by arguments based on the discovered principles. But if they must have particular and complete experience of their conclusions, then it is necessary that they should have it by the aid of this noble science. It is true, indeed, that mathematics has universal experiences concerning its conclusions in figuring and numbering, which are applied likewise to all the sciences and to this experimental science, because no science can be known without mathematics. But if we turn our attention to the experiences that are particular and complete and certified wholly in their own discipline, it is necessary to go by way of the principles of this science which is called experimental.

The other prerogatives of experimental science, besides this first one of confirming the conclusions of deductive reasoning in existing sciences, as, for example, in optics, were, secondly, to add to existing sciences new knowledge at which they could not arrive by deduction, and thirdly, to create entirely new departments of science. By virtue of these two prerogatives the experimenter was able tomake a purely empirical discovery of the nature of things. Of the second prerogative Bacon said:

This mistress of the speculative sciences alone is able to give us important truths within the confines of the other sciences, which those sciences can learn in no other way. Hence these truths are not connected with the discussion of principles but are wholly outside of these, although they are within the confines of these sciences, since they are neither conclusions nor principles.… The man without experience must not seek a reason in order that he may first understand, for he will never have this reason except after experiment.… For if a man is without experience that a magnet attracts iron, and has not heard from others that it attracts, he will never discover this fact before an experiment.… Mathematical science can easily produce the spherical astrolabe, on which all astronomical phenomena necessary for man may be described, according to precise longitudes and latitudes [as in the device described by Ptolemy in the Almagest, viii]. But that this body, so made, should move naturally with the daily motion is not within the power of mathematical science. But the trained experimenter can consider the ways of this motion.

Other examples of the exercise of the second prerogative were seen in medicine and in alchemy. The third prerogative was exercised outside the bounds of existing sciences, as in the investigation of natural wonders and prognostications of the future.

The inductive process of the discovery, as well as the verification and falsification of principles or theories, Roger Bacon explained fully and clearly in the example he gave to illustrate the first prerogative, though he did not included discovery in the special meaning he gave to the phrase 'scientia experimentalis' in the passage concerning this prerogative quoted above. But, before discussing this 'example of the rainbow and of the phenomena connected with it', it would be well to turn briefly to his ideas about the use of mathematics and of optics.

Mathematics, Roger Bacon said, was the 'door and key' 'of the sciences and things of this world' and gave certain knowledge of them. In the first place 'all categories depend on a knowledge of quantity, concerning which mathematics treats, and therefore the whole excellence of logic depends on mathematics'. For 'the categories of when and where are related to quantity, for when pertains to time and where arises from place; the category of condition(habitus) cannot be known without the category of where, as Averroës teaches in the fifth book of the Metaphysics; the greater part, moreover, of the category of quality contains affections and properties of quantities, because all things that are in the fourth class of quality are called qualities in quantities… whatever, moreover, is worthy of consideration in the category of relation is the property of quantity, such as proportions and proportionalities, and geometrical, arithmetical, and musical means and the kinds of greater and lesser inequality.' This being the case it was plain that 'mathematics is prior to the other sciences', and since 'in mathematics only, as Averroës says in the first book of the Physics…, things known to us and in nature or absolutely are the same', the greatest certainty was possible in mathematics. 'In mathematics only are there the most convincing demonstrations through a necessary cause.' 'Wherefore it is evident that if, in the other sciences, we want to come to certitude without doubt and to truth without error, we must place the foundations of knowledge in mathematics.' 'Robert, Bishop of Lincoln and Brother Adam of Marsh' had followed this method and 'if anyone should descend to the particular by applying the power of mathematics to the separate sciences, he would see that nothing magnificent in them can be known without mathematics.'

As an example of the use of mathematics in making known 'the things of this world' Roger Bacon gave astronomy, which 'considers the quantities of all things that are included among the celestial and all things which are reduced to quantity'. He said that 'by instruments suitable to them and by tables and canons' the movements of the celestial bodies and other phenomena in the heavens and in the air might be measured and reduced to rules on which predictions might be based. In fact he carried on the work of Grosseteste towards the reform of the calendar, making use of Grosseteste's Compotus and also sharing his hesitation between the Aristotelian and Ptolemaic astronomical systems.

The special reason why Bacon held that 'in the things of this world, as regards their efficient and generating causes, nothing can be known without the power of geometry', and that 'it is necessary to verify the matter of the world by demonstrations set forth in geometrical lines', was that he accepted Grosseteste's theory of the 'multiplication of species' or power as the basis of all natural operations andthe Neoplatonic theory of a 'common corporeity' as the first form giving dimensions to all material substances. Like Grosseteste, he held that the 'multiplication of species' was the efficient cause of every occurrence in the universe, whether in the celestial or terrestrial region, whether in matter or in sense, and whether originating from inanimate things or from the soul. And, he said, 'the force of the efficient cause and of the matter cannot be known without the great power of mathematics', for 'Every multiplication is either with respect to lines, or angles, or figures'.

In discussing the 'multiplication of species' Roger Bacon based conclusions on the same basic metaphysical principles that Grosseteste had used. The principle of the uniformity of nature he expressed as follows: 'the effects… will be similar to those in the past, since if we assume a cause the effect is taken for granted', and 'those which are of similar essence have similar operations'. The principle of economy he expressed in Grosseteste's own words: 'Aristotle says in the fifth book of the Metaphysics that nature works in the shortest way possible, and the straight line is the shortest way of all.' The type of such 'species' was visible light and therefore, like Grosseteste, he made a particular study of geometrical optics, through which he held that it was possible to obtain experimental knowledge of the laws of the operation of these species, which laws were the basis of all natural explanation.

Of the mode of propagation of 'species' Roger Bacon gave an account which resumed and extended some of the essential features of Grosseteste's 'wave' theory. He asserted first that for propagation between two points to occur at all the intervening medium must be a plenum: no propagation could pass through an absolute void.

Democritus thought that an eye on the earth could see an ant in the heavens.… But we must here state that we should not see anything if there were a vacuum. But this would not be due to some nature hindering species, and resisting it, but because of the lack of a nature suitable for the multiplication of species, for species is a natural thing, and therefore needs a natural medium; but in a vacuum nature does not exist. For vacuum rightly conceived of is merely a mathematical quantity extended in the three dimensions, existing per se without heat and cold, soft and hard, rare and dense, and without any natural quality, merely occupying space, as the philosophers maintained before Aristotle, not only within the heavens, but beyond.

He then went on to argue that the propagation was not instantaneous but took time. Alhazen had brought various arguments against Alkindi's attempt in De Aspectibus to prove that 'the ray passes through in a wholly indivisible instant'. After a long discussion based on such considerations as that 'a finite force cannot produce any result in an instant, wherefore it must require time', and that since 'the species of a corporeal thing has a really corporeal existence in a medium, and is a real corporeal thing, as was previously shown, it must of necessity be dimensional, and therefore fitted to the dimensions of the medium', Bacon concluded: 'It remains, then, that light is multiplied in time, and likewise all species of a visible thing and of vision. But nevertheless the multiplication does not occupy a sensible time and one perceptible by vision, but an imperceptible one, since anyone has experience that he himself does not perceive the time in which light travels from east to west.'

This multiplication of species through a medium, he continued, was not a flow of body like water but a kind of pulse propagated from part to part. In this, light was analogous to sound.

For sound is produced because parts of the object struck go out of their natural position, where there follows a trembling of the parts in every direction along with some rarefaction, because the motion of rarefaction is from the centre to the circumference, and just as there is generated the first sound with the first tremor, so is there a second sound with the second tremor in a second portion of the air, and a third sound with the third tremor in a third portion of the air, and so on.

[With light the species] forms a likeness to itself in the second position of the air, and so on. Therefore it is not a motion as regards place, but is a propagation multiplied through the different parts of the medium; nor is it a body which is there generated, but a corporeal form, without, however, dimensions per se, but it is produced subject to the dimensions of the air; and it is not produced by a flow from a luminous body, but by a renewing from the potency of the matter of the air.… As regards Aristotle's statement that there is a difference between the transmission of light and that of the other sensory impressions,… sound has the motion of the displacement of the parts of the body struck from its natural position, and the motion of the following tremor, and the motion of rarefaction in every direction, as was stated before, and as is evident from the second book of De Anima;and… in the multiplication of sound a three-fold temporal succession takes place, no one of which is present in the multiplication of light.… However, the multiplication of both as regards itself is successive and requires time. Likewise, in the case of odour the transmission is quite different from that of light, and yet the species of both will require time for transmission, for in odour there is a minute evaporation of vapour, which is, in fact, a body diffused in the air to the sense besides the species, which is similarly produced.… But in vision nothing is found except a succession of the multiplication. The fact that there is a difference in the transmission of light, sound, and odour can be set forth in another way, for light travels far more quickly in the air than the other two. We note in the case of one at a distance striking with a hammer or a staff that we see the stroke delivered before we hear the sound produced. For we perceive with our vision a second stroke, before the sound of the first stroke reaches the hearing. The same is true of a flash of lightning, which we see before we hear the sound of the thunder, although the sound is produced in the cloud before the flash, because the flash is produced in the cloud from the bursting of the cloud by the kindled vapour.

In the details of his researches into optics and cognate sciences Roger Bacon made use of a number of Grosseteste's ideas, though his work was more mature because of the new sources available to him and he usually added something original of his own. Besides Grosseteste, his chief sources in optics were Aristotle, Euclid, and pseudo-Euclid, Ptolemy, Diocles (Tideus), Alhazen, Alkindi, Avicenna, and Averroëes. He took over and extended Grosseteste's explanation of the variation in the strength of rays according to direction, and according to the distance from the radiating source; examples, respectively, of the multiplication of species 'according to lines and angles' and 'according to figures'. The results he used in an interesting discussion of which the object was to

verify the fact that on the surface of the lens of the eye, although it be small, the distinction of any visible object whatsoever can be made by means of the arrangement of the species coming from such objects, since the species of a thing, whatever be its size, can be arranged in order in a very small space, because there are as many parts in a very small body as there are in a very large one, since every body and every quantity is infinitely divisible, as all philosophy pro-claims. Aristotle proves in the sixth book of the Physics that there is no division of a quantity into indivisibles, nor is a quantity composed of indivisibles, and therefore there are as many parts in a grain of millet as in the diameter of the world.

He showed then that it was possible to draw an infinite number of lines from the base of a triangle to the point at its apex.

Roger Bacon made use also of Grosseteste's explanation of the tides. He incorporated a section of De Natura Locorum in the section of the Opus Maius dealing with the effects of rays on climate, and he seems to be referring to De Iride in the remark in the Opus Tertium that 'homines habentes oculos profundos longius vident'. He took over Grosseteste's theory of heat and expanded his remarks about the internal strain between the parts of a body, which produced an intrinsic resistance to movement in falling bodies because each part prevented those lateral to it from going straight to the centre of the earth. 'This conclusion, that a heavy body receives a strain in its own natural motion, is proved by cause and effect', he said, and motion under strain generated heat. He made use of Grosseteste's theory of double refraction to explain the operation of a spherical (and hemispherical) lens or burning-glass, adding: 'instruments can be made so that we may sensibly see propagations of this kind; but until we have instruments we can prove this by natural effect without contradiction.… Let us take a hemisphere of crystal or a glass vessel, the lower part of which is round and full of water.'This, he said, should be held in the rays of the sun, as Grosseteste described in De Natura Locorum. He took up also Grosseteste's suggestion as to the possibilities of using lenses for magnifying small objects, and he made experiments with plano-convex lenses while trying to use the laws of refraction to improve vision, a practical object such as he held to be the final justification of all theoretical science.

If anyone examine letters or other small objects through the medium of a crystal or glass or some other transparent body placed above the letters, and if it be shaped like the lesser segment of a sphere with the convex side towards the eye, and the eye is in the air, he will see the letters much better and they will appear larger to him. For in accordance with the truth of the fifth rule regarding a spherical medium beneath which the object is placed, the centre being beyond the object and the convexity towards the eye, all conditions are favourable for magnification, for the angle in which it is seen is greater, the image is greater, and the position of the image is nearer, because the object is between the eye and the centre. For this reason this instrument is useful to old people and people with weak eyes, for they can see any letter however small if magnified enough.

To the eye and its functioning in vision Roger Bacon paid particular attention because, as he said, 'by means of it we search out certain experimental knowledge of all things that are in the heavens and in the earth'. His account of vision was one of the most important written during the Middle Ages and it became a point of departure for seventeenth-century work. Bacon's chief contribution was to try to explain the operation of the eye, of which his account was based largely on the writings of Alhazen and Avicenna, by means of the theory of 'multiplication of species'. Distinguishing like Grosseteste between the psychological act of vision which went forth from the eye, and the physical light which went from the visible object to the eye, he asserted that both the extramitted species of vision and the intramitted species of light from the visible object were necessary.

The reason for this assertion is that everything in nature completes its action through its own force and species alone… as, for example, fire by its own force dries and consumes and does many things. Therefore vision must perform the act of seeing by its own force. But the act of seeing is the perception of a visible object at a distance, and therefore vision perceives what is visible by its own force multiplied to the object. Moreover, the species of the things of the world are not fitted by nature to effect the complete act of vision at once, because of its nobleness. Hence these must be aided and excited by the species of the eye, which travels in the locality of the visual pyramid, and changes the medium and ennobles it, and renders it analogous to vision, and so prepares the passage of the species itself of the visible object.… Concerning the multiplication of this species, moreover, we are to understand that it lies in the same place as the species of the thing seen, between the sight and the thing seen, and takes place along the pyramid whose vertex is in the eye and base in the thing seen. And as the species of an object in the same medium travels in a straight path and is refracted indifferent ways when it meets a medium of another transparency, and is reflected when it meets the obstacle of a dense body; so is it also true of the species of vision that it travels altogether along the path of the species itself of the visible object.

To show how the eye focused the species of light entering it he described first the anatomical arrangement of its parts. Following Avicenna he said that the eye had three coats and three humours. The inner coat consisted of two parts, the rete or retina, an expansion of the nerve forming a concave net 'supplied with veins, arteries and slender nerves' and acting as a conveyor of nourishment; and out-side this a second thicker part called the uvea. Outside the uvea were the cornea, which was transparent where it covered the opening of the pupil, and the consolidativa or conjunctiva. Inside the inner coat were the three humours, and so, for light entering the pupil: 'There will then be the cornea, the humor albigineus, the humor glacialis [lens], and the humor vitreus, and the extremity of the nerve, so that the species of things will pass through the medium of them all to the brain.… The crystalline humour [lens] is called the pupil, and in it is the visual power.

The theory of vision Bacon described was essentially that of Alhazen and in fact misunderstanding of the functions of the lens and retina remained the chief stumbling block to the formulation of an adequate theory of vision until the end of the sixteenth century. Of the theory that the lens was the only sensitive part of the eye Bacon wrote, using what became known as the method of agreement and difference, as Alhazen had done: 'For if it is injured, even though the other parts are whole, vision is destroyed, and if it is unharmed and injury happens to the others, provided they retain their transparent quality, vision is not destroyed.' But, in another passage inspired by one of Alhazen's chapters, he stressed the qualification.

that vision is not completed in the eye, but in the nerve… for two different species come to the eyes and… in two eyes there are different judgements.… Therefore there must be something sentient besides the eyes, in which vision is completed and of which the eyes are the instruments that give it the visible species. This is the common nerve in the surface of the brain, where the two nerves coming from the two parts of the anterior brain meet, and after meeting are divided and extend to the eyes.… But it is necessary that the two species coming from the eyes should meet at one place in the common nerve, and that one of these should be more intense and fuller than the other. For naturally the two forms of the same species mingle in the same matter and in the same place, and therefore are not distinguished, but become one form after they come to one place, and then, since the judging faculty is single and the species single, a single judgement is made regarding the object. A proof of this is the fact that when the species do not come from the two eyes to one place in the common nerve, one object is seen as two. This is evident when the natural position of the eyes is changed, as happens if the finger is placed below one of the eyes or if the eye is twisted somewhat from its place; both species do not then come to one place in the common nerve, and one object is seen as two.

In another passage Roger Bacon tried to show how the 'visible species' were focused on the end of the optic nerve without producing an inverted image. In common with all optical writers before Kepler he failed to understand that such an image was compatible with normal vision.

If the rays of the visual pyramid meet at the centre of the anterior glacialis [lens], they must be mutually divided and what was right would become left and the reverse, and what was above would be below.… In order, therefore, that this error may be avoided and the species of the right part may pass on its own side, and the left to its side, and so too of other positions, there must be something else between the anterior of the glacialis and its centre to prevent a meeting of this kind. Therefore Nature has contrived to place the vitrous humour before the centre of the glacialis, which has a different transparency and a different centre, so that refraction takes place in it, in order that the rays of the pyramid may be diverted from meeting in the centre of the anterior glacialis. Since, therefore, all rays of the radiant pyramid except the axis… are falling at oblique angles on the vitreous humour… all those rays must be refracted on its surface.… Since, moreover, the vitreous humour is denser than the anterior glacialis, it follows, therefore, that refraction takes place between the straight path and the perpendicular drawn at the point of refraction, as has been shown in the multiplication of species.… Thus the right species will always go according to its own direction until it comes to a point of the common nerve… and will not go to the left.… The same is true of the species coming from all other parts.

Roger Bacon was mistaken in thinking that the vitreous humour had a higher refractive index than the lens, and in other respects his theory of vision was far from correct. Nevertheless, his attempt to solve the problem of how the image was formed behind the lens was a step in the right direction. He thought that the nerve was 'filled with a similar vitreous humour as far as the common nerve' so that the 'species' travelled along it without refraction, though caused by 'the power of the soul's force (virtutis)… to follow the tortuosity of the nerve, so that it flows along a tortuous line, not along a straight one, as it does in the inanimate bodies of the world'. In the common nerve the judgements of the 'visual faculty' (virtus visiva) were completed, so that it was the seat of 'ultimate perception' in vision. The other special senses were analogously accommodated. Where more than one special sense was involved the 'ultimate perception' occurred at a deeper level, in 'the common sense (sensus communis) in the anterior part of the brain'.

His knowledge of optics Roger Bacon used in the experimental-mathematical investigation of the cause of the rainbow which he gave in Part VI of the Opus Maius as an example of his method. His procedure, in fact, followed the essential principles of Grosseteste's methods of combined resolution and composition and of falsification, and it represents the first major advance made in the experimental method after Grosseteste. He began by collecting instances of phenomena similar to the rainbow, both as to the colours and the bow-like shape. He said:

The experimenter, then, should first examine visible objects in order that he may find colours arranged as in the phenomenon mentioned above and also the same figure. For let him take hexagonal stones from Ireland and from India, which are called iris stones in Solinus on the Wonders of the World, and let him hold these in a solar ray falling through the window, so that he may find in the shadow near the ray all the colours of the rainbow, arranged as in it. And further let the same experimenter turn to a somewhat dark place and apply the stone to one of his eyes which is almost closed, and he will see the colours of the rainbow clearly arranged just as in the bow. And since many employing these stones think that the phenomenon is due to the special virtue of those stones and to their hexagonal shape, therefore let the experimenter proceed farther, and he will find this same peculiarity in crystalline stones correctly shaped, and in other transparent stones. Moreover, he will find this not only in white stones like the Irish crystals, but also in black ones, as is evident in the dark crystal and in all stones of similar transparency. He will find it besides in crystals of a shape differing from the hexagonal, provided they have a roughened surface, like the Irish crystals, neither altogether smooth, nor rougher than they are. Nature produces some that have surfaces like the Irish crystals. For a difference in the corrugations causes a difference in the colours. And further let him observe rowers, and in the drops falling from the raised oars he finds the same colours when the solar rays penetrate drops of this kind. The same phenomenon is seen in water falling from the wheels of a mill; and likewise when one sees on a summer's morning the drops of dew on the grass in a meadow or field, he will observe the colours. Likewise when it is raining, if he stands in a dark place, and the rays beyond it pass through the falling rain, the colours will appear in the shadow nearby; and frequently at night colours appear round a candle. Moreover, if a man in summer, when he rises from sleep and has his eyes only partly open, suddenly looks at a hole through which a ray of the sun enters, he will see colours. Moreover, if seated out of the sun he holds his cap beyond his eyes, he will see colours; and similarly if he closes an eye the same thing happens in the shade of his eyebrows; and again the same phenomenon appears through a glass vessel filled with water and placed in the sun's rays. Or similarly if someone having water in his mouth sprinkles it vigorously into the rays and stands at the side of the rays. So, too, if rays in the required position pass through an oil lamp hanging in the air so that the light falls on the surface of the oil, colours will be produced. Thus in an infinite number of ways colours of this kind appear, which the diligent experimenter knows how to discover.

In a similar way also he will be able to test the shape in which the colours are disposed. For by means of the crystalline stone and substances of this kind he will find the shape straight. By means of the eyelids and eyebrows and by many other means, and also by means of holes in rags, he will discover whole circles coloured. Similarly, in a place where the dewfall is plentiful and sufficient to take the whole circle, and if the place where the circle of the rainbow should be is dark proportionately, because the bow does not appear in the light part, then the circle will be complete. Similarly, whole circles appear frequently around candles, as Aristotle states and we ourselves experience.

Since, moreover, we find colours and various figures similar to the phenomena in the air, namely, of the iris, halo, and mock-suns, we are encouraged and greatly stimulated to grasp the truth in those phenomena that occur in the heavens.

From an examination of these instances Bacon tried to reach a 'common nature' uniting the rainbow and similar phenomena, and in the course of his argument he considered several different theories and eliminated those contradicted by observation. To explain the variation in the altitude of rainbows he took over Aristotle's theory that the rainbow formed part of the circumference of the base of a cone of which the apex was at the sun and the axis passed through the observer's eye to the centre of the bow, and he confirmed this by showing by measurements with the astrolabe that the sun, the observer's eye, and the centre of the bow were always in a straight line. This explained why the altitude of the bow varied at different latitudes and different times of year, and why a complete circle could be seen only when the base of the cone was elevated above the surface of the earth, as with rainbows in sprays.

The experimenter, therefore, taking the altitude of the sun and of the rainbow above the horizon will find that the final altitude at which the rainbow can appear above the horizon is 42 degrees, and this is the maximum elevation of the rainbow.… And the rainbow reaches this maximum elevation when the sun is on the horizon, namely, at sunrise and sunset.

In the latitude of Paris, he said, 'the altitude of the sun at noon of the equinox is 41 degrees and 12 minutes', and therefore in the summer, when the altitude of the sun is greater than 42 degrees, no rainbow can appear at noon. He discussed in some detail the times of year when rainbows could not appear in Scotland, Jerusalem, and other places.

Going on to discuss 'whether the bow is caused by incident rays or by reflection or by refraction, and whether it is an image of the sun… and whether there are real colours in the cloud itself', he said: 'to understand these matters we must employ definite experiments'. He pointed out that each observer saw a different bow which moved when he did in relation to trees and other fixed objects, whether he moved parallel to, towards, or away from the bow. There were, he said, as many bows as observers, for each observer saw his bow follow his own movement, his shadow bisecting its arc. Therefore the rainbow could not be seen by 'incident' (i. e. direct) rays, for if it were it would appear fixed in one place like the white and black patches on clouds.

Similarly, when a colour is produced by incident rays through a crystalline stone, refraction takes place in it, but the same colour in the same position is seen by different observers.… Moreover, the image of an object seen by refraction does not follow the observer if he recedes, nor does it recede if he approaches, nor does it move in a direction parallel to him; which is evident when we look at a fish at rest in water, or a stick fixed in it, or the sun or moon through the medium of vapours, or letters through a crystal or glass.

Therefore, since there were only three kinds of 'principal rays' (direct, refracted and reflected), and since 'accidental rays… do not change their position unless caused by reflection', the rainbow must be seen by reflected rays. 'All the raindrops have the nature of a mirror', and things seen in a mirror moved when the observer moved, just as the rainbow did. 'There are, then, raindrops of small size in infinite number, and reflection takes place in every direction as from a spherical mirror.' Yet the rainbow could not be an image of the sun produced by such reflection, as Seneca had suggested, because spherical mirrors distorted the shape and changed the size and colour of objects seen in them.

Of the colours seen in the rainbow and in crystals, Roger Bacon said:

If it be said that solar rays passing through a crystal produce real and fixed colours, which produce a species and have objective reality, we must reply that the phenomena are different. The observer alone produces the bow, nor is there anything present except reflection. In the case of the crystal, however, there is a natural cause, namely, the ray and the corrugated stone, which has great diversity of surface, so that according to the angle at which the light falls a diversity of colours result. And viewing them does [not] cause the colours to be present here, for there is colour before it is seen here, and it is seen by different people in the same place. But in the case of the bow the phenomenon is the result of vision, and therefore can have no reality but merely appearance.

The theory that Albertus Magnus had advanced, that the colours of the rainbow were due to differences in density of cloud, Roger Bacon rejected on the grounds that there were no such differences in crystals, or in sprays or dew on the grass, where, nevertheless, similar colours were seen. Real colours such as those seen in hexagonal crystals he attributed to mixtures of white and black, as explained by Aristotle in De Sensu et Sensibili. Of the colours of the rainbow he said: 'We need give only the cause of the appearance.' 'It is thought by scientists that these colours are caused by the humours and colours of the eye, for these colours exist merely in appearance.'

Concerning the shape of the rainbow, Bacon considered and rejected two earlier theories. First, he said that it could not be produced by the raindrops themselves falling in a cone, for the circular shape appeared in irregular sprays. Secondly, he attacked Grosseteste's theory that the bow was produced by three separate refractions through successively denser layers of moist atmosphere. He said that only one refraction could take place in sprays, yet the same shape was formed as seen in the sky. Moreover, Grosseteste's statement that the refracted rays would spread out, 'not into a round pyramid [i. e. cone], but into a figure like the curved surface of a round pyramid', seemed to Bacon to contradict the law of refraction, according to which the rays would form a regular cone. Nor could the curvature be produced by the moisture, for according to Grosseteste this was not of such a form but was 'a rounded mass of conical form'.

'Another explanation must therefore be sought; and it can be stated that the bow must be in the form of a circular arc.' For the colours of the rainbow did not shift with varying incidence of light like those on the dove's neck, but 'the same colour in one circle of the bow appears from one end to the other, and therefore all parts must have the same position with respect to the solar ray and the eye'. This condition and the appearance of the rainbow would be satisfied if the rainbow were a circle with its centre on the line joining the sun and the eye. He concluded:

every where [where there are raindrops] there are conditions suitable for the appearance of the bow, but as an actual fact the bow appears only in raindrops from which there is reflection to the eye; because there is merely the appearance of colours arising from the imagination and deception of the vision.… A reflection comes from every drop at the same time, while the eye is in one position, because of the equality of the angles of incidence and reflection.

Bacon's understanding of the part played by individual raindrops in the formation of the rainbow was a real advance, in spite of his rejection of refraction. He extended his knowledge of optics to try to explain halos, mock-suns, and other similar phenomena. His explanation of the halo is interesting because it was based on the explicit assumption that the sun's rays were parallel. He said in the Opus Maius that the halo was caused by rays going out from the sun 'like a cylinder in shape' and becoming refracted on passing through a spherical mass of vapour in the atmosphere between the sun and the eye, so as to go to the eye in a cone. The reason for the shape of the halo was that 'All the rays falling on one circular path round that axis [joining the sun and the eye] are refracted at equal angles, because all the angles of incidence are equal'. But, he continued, 'just as many experiments are needed to determine the nature of the rainbow both in regard to its colour and its shape, so too are they required in this investigation'.

Taking up the same subject in the Opus Tertium, he said that each eye saw a different halo, which moved as it did. In this work he attributed the refraction of the sunlight to individual water-drops. He pointed out also that colours seen in a halo were in the reverse order to those seen in the primary rainbow, and that measurements with an astrolabe showed that the diameter of the halo subtended at the eye of the observer an angle of 42 degrees, the same angle as that subtended by the radius of the rainbow. The sentiments with which he concluded his account in the Opus Maius of the first prerogative of experimental science are a worthy expression of the ideals of the experimental method by one of its founders:

Hence reasoning does not attest these matters, but experiments on a large scale made with instruments and by various necessary means are required. Therefore no discussion can give an adequate explanation in these matters, for the whole subject is dependent on experiment. For this reason I do not think that in this matter I have grasped the whole truth, because I have not yet made all the experiments that are necessary, and because in this work I am proceeding by the method of persuasion and of demonstration of what is required in the study of science, and not by the method of compiling what has been written on this subject. Therefore it does not devolve on me to give at this time an attestation impossible for me, but to treat the subject in the form of a plea for the study of science.