This collection of first-rate essays aims to provide an accurate scholarly assessment of the relationship between the Roman Catholic Church and Galileo. In , Pope John Paul II established a commission to inquire into the Church's treatment of Galileo "in loyal recognition of wrongs, from whatever side they came," hoping this way to "dispel the mistrust.

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This volume attempts what the Commission failed to provide—a historically accurate, scholarly, and balanced account of Galileo and his difficult relationship with the Roman Catholic Church. Contributors provide careful analyses of the interactions of the Church and Galileo over the thirty years between and his death in They also explore the attitudes of theologians to the Copernican innovation prior to Galileo's entry into the fray; survey the political landscape within which he lived; assess the effectiveness or otherwise of censorship of his work; and provide an analysis and occasional critique of the Church's later responses to the Galileo controversy.

This collection of first-rate essays aims to provide an accurate scholarly assessment of the relationship between the Roman Catholic Church and Galileo. In , Pope John Paul II established a commission to inquire into the Church's treatment of Galileo "in loyal recognition of wrongs, from whatever side they came," hoping this way to "dispel the mistrust. This volume attempts what the Commission failed to provide—a historically accurate, scholarly, and balanced account of Galileo and his difficult relationship with the Roman Catholic Church.

Contributors provide careful analyses of the interactions of the Church and Galileo over the thirty years between and his death in They also explore the attitudes of theologians to the Copernican innovation prior to Galileo's entry into the fray; survey the political landscape within which he lived; assess the effectiveness or otherwise of censorship of his work; and provide an analysis and occasional critique of the Church's later responses to the Galileo controversy.

And here I wish you to observe that after dividing and resolving a line into a finite number of parts, that is, into a number which can be counted, it Edition: NatlEd; Page: [ 72 ] is not possible to arrange them again into a greater length than that which they occupied when they formed a continuum [ continuate ] and were connected without the interposition of as many empty spaces. But if we consider the line resolved into an infinite number of infinitely small and indivisible parts, we shall be able to conceive the line extended indefinitely by the interposition, not of a finite, but of an infinite number of infinitely small indivisible empty spaces.

Now this which has been said concerning simple lines must be understood to hold also in the case of surfaces and solid bodies, it being assumed that they are made up of an infinite, not a finite, number of atoms. Such a body once divided into a finite number of parts it is impossible to reassemble them so as to occupy more space than before unless we interpose a finite number of empty spaces, that is to say, spaces free from the substance of which the solid is made. But if we imagine the body, by some extreme and final analysis, resolved into its primary elements, infinite in number, then we shall be able to think of them as indefinitely extended in space, not by the interposition of a finite, but of an infinite number of empty spaces.

Thus one can easily imagine a small ball of gold expanded into a very large space without the introduction of a finite number of empty spaces, always provided the gold is made up of an infinite number of indivisible parts. It seems to me that you are travelling along toward those vacua advocated by a certain ancient philosopher.

I noticed, and not without indignation, the rancor of this ill-natured opponent; further references to these affairs I omit, not only as a matter of good form, but also because I know how unpleasant they are to the good tempered and well ordered mind of one so religious and pious, so orthodox and God-fearing as you. But to return to our subject, your previous discourse leaves with me many difficulties which I am unable to solve. First among these is that, if the circumferences of the two circles are equal to the two straight lines, CE and BF, the latter considered as a continuum, the former as interrupted with an infinity of empty points, I do not see how it is possible to say that the line AD described by the center, and made up of an infinity of points, is equal to this center which is a single point.

## Galileo : Decisive Innovator

Besides, this building up of lines out of points, divisibles out of indivisibles, and finites out of infinites, offers me an obstacle difficult to avoid; and the necessity of introducing a vacuum, so conclusively refuted by Aristotle, presents the same difficulty. These difficulties are real; and they are not the only ones. But let us remember that we are dealing with infinities and indivisibles, both of which transcend our finite understanding, the former on account of their magnitude, the latter because of their smallness.

In spite of this, men cannot refrain from discussing them, even though it must be done in a round-about way. Therefore I also should like to take the liberty to present some of my ideas which, though not necessarily convincing, would, on account of their novelty, at least, prove somewhat startling. But such a diversion might perhaps carry us too far away from the subject under discussion and might therefore appear to you inopportune and not very pleasing.

Pray let us enjoy the advantages and privileges which come from conversation between friends, especially upon subjects freely chosen and not forced upon us, a matter vastly different from dealing with dead books which give rise to many doubts but remove none.

Share with us, therefore, the thoughts Edition: current; Page: [ 27 ] which our discussion has suggested to you; for since we are free from urgent business there will be abundant time to pursue the topics already mentioned; and in particular the objections raised by Simplicio ought not in any wise to be neglected. Granted, since you so desire. The first question was, How can a single point be equal to a line? Since I cannot do more at present I shall attempt to remove, or at least diminish, one improbability by introducing a similar or a greater one, just as sometimes a wonder is diminished by a miracle.

And this I shall do by showing you two equal surfaces, together with two equal solids located upon these same surfaces as bases, all four of which diminish continuously and uniformly in such a way that their remainders always preserve equality among themselves, and finally both the surfaces and the solids terminate their previous constant equality by degenerating, the one solid and the one surface into a very long line, the other solid and the other surface into a single point; that is, the latter to one point, the former to an infinite number of points.

This proposition appears to me wonderful, indeed; but let us hear the explanation and demonstration. Since the proof is purely geometrical we shall need a figure. Imagine the radius CF to be drawn perpendicular to either of the lines AB or DE, and the entire figure to rotate about this radius as an axis. Besides this we shall prove that the base of the cone, i. Here we have the miracle mentioned above; as the cutting plane approaches the line AB the portions of the solids cut off are always equal, so also the areas of their bases.

And as the cutting plane comes near the top, the two solids always equal as well as their bases areas which are also equal finally vanish, one pair of them degenerating into the circumference of a circle, the other into a single point, namely, the upper edge of the bowl and the apex of the cone. Now, since as these solids diminish equality is maintained between them up to the very last, we are justified in saying that, at the extreme and final end of this diminution, they are still equal and that one is not infinitely greater than the other.

## Io Voyager and Galileo Global Mosaics | USGS Astrogeology Science Center

It appears therefore that we may equate the circumference of a large circle to a single point. And this which is true of the solids is true also of the surfaces which Edition: current; Page: [ 29 ] form their bases; for these also preserve equality between themselves throughout their diminution and in the end vanish, the one into the circumference of a circle, the other into a single point. Shall we not then call them equal seeing that they are the last traces and remnants of equal magnitudes?

Note also that, even if these vessels were large enough to contain immense celestial hemispheres, both their upper edges and the apexes of the cones therein contained would always remain equal and would vanish, the former into circles having the dimensions of the largest celestial orbits, the latter into single points. Hence in conformity with the preceding we may say that all circumferences of circles, however different, are equal to each other, and are each equal to a single point. This presentation strikes me as so clever and novel that, even if I were able, I would not be willing to oppose it; for to deface so beautiful a structure by a blunt pedantic attack would be nothing short of sinful.

But for our complete satisfaction Edition: NatlEd; Page: [ 76 ] pray give us this geometrical proof that there is always equality between these solids and between their bases; for it cannot, I think, fail to be very ingenious, seeing how subtle is the philosophical argument based upon this result. The demonstration is both short and easy. And, since the areas of circles are to each other as the squares of their diameters, it follows that the area of the circle whose diameter is GN is equal to the sum of the areas of circles having diameters IO and HL, so that if we remove the common area of the circle having IO for diameter the remaining area of the circle GN will be equal to the area of the circle whose diameter is HL.

So much for the first part. The demonstration is ingenious and the inferences drawn from it are remarkable. And now let us hear something concerning the other difficulty raised by Simplicio, if you have anything special to say, which, however, seems to me hardly possible, since the matter has already been so thoroughly discussed. But I do have something special to say, and will first of all repeat what I said a little while ago, namely, that infinity and indivisibility are in their very nature incomprehensible to us; imagine then what they are when combined. Yet if Edition: NatlEd; Page: [ 77 ] we wish to build up a line out of indivisible points, we must take an infinite number of them, and are, therefore, bound to understand both the infinite and the indivisible at the same time.

Many ideas have passed through my mind concerning this subject, some of which, possibly the more important, I may not be able to recall on the spur of the moment; but in the course of our discussion it may happen that I shall awaken in you, and especially in Simplicio, objections and difficulties which in turn will bring to memory that which, without such stimulus, would have lain dormant in my mind. Allow me therefore the customary liberty of introducing some of our human fancies, for indeed we may so call them in comparison with supernatural truth which furnishes the one true and safe recourse for decision in our discussions and which is an infallible guide in the dark and dubious paths of thought.

Thus if two indivisibles, say two points, can be united to form a quantity, say a divisible line, then an even more divisible line might be formed by the union of three, five, seven, or any other odd number of points. Since however these lines can be cut into two equal parts, it becomes possible to cut the indivisible which lies exactly in the middle of the line. In answer to this and other objections of the same type we reply that a divisible magnitude cannot be constructed out of two or ten or a hundred or a thousand indivisibles, but requires an infinite number of them.

Here a difficulty presents itself which appears to me insoluble. Since it is clear that we may have one line greater than another, each containing an infinite number of points, we are forced to admit that, within one and the same class, we may have something greater than infinity, because the infinity of points in the long line is greater than the infinity of points in the short line.

This assigning to an infinite quantity a value greater than infinity is quite beyond my comprehension. This is one of the difficulties which arise when we attempt, with our finite minds, to discuss the infinite, assigning to it those properties which we give to the finite and limited; but Edition: NatlEd; Page: [ 78 ] this I think is wrong, for we cannot speak of infinite quantities as being the one greater or less than or equal to another.

To prove this I have in mind an argument which, for the sake of clearness, I shall put in the form of questions to Simplicio who raised this difficulty. I take it for granted that you know which of the numbers are squares and which are not. I am quite aware that a squared number is one which results from the multiplication of another number by itself; thus 4, 9, etc. Very well; and you also know that just as the products are called squares so the factors are called sides or roots; while on the other hand those numbers which do not consist of two equal factors are not squares.

Therefore if I assert that all numbers, including both squares and non-squares, are more than the squares alone, I shall speak the truth, shall I not? If I should ask further how many squares there are one might reply truly that there are as many as the corresponding number of roots, since every square has its own root and every root its own square, while no square has more than one root and no root more than one square.

But if I inquire how many roots there are, it cannot be denied that there are as many as there are numbers because every number is a root of some square.

### Product details

This being granted we must say that there are as many squares as there are numbers because they are just as numerous as their roots, and all the numbers are roots. Yet at the outset we said there are many more numbers than squares, since the larger portion of them are not squares. Not only so, but the proportionate number of squares diminishes as we pass to larger numbers. When therefore Simplicio introduces several lines of different lengths and asks me how it is possible that the longer ones do not contain more points than the shorter, I answer him that one line does not contain more or less or just as many points as another, but that each line contains an infinite number.

Or if I had replied to him that the points in one line were equal in number to the squares; in another, greater than the totality of numbers; and in the little one, as many as the number of cubes, might I not, indeed, have satisfied him by thus placing more points in one line than in another and yet maintaining an infinite number in each? So much for the first difficulty. Pray stop a moment and let me add to what has already been said an idea which just occurs to me.

### Program Description

If the preceding be true, it seems to me impossible to say either that one infinite number is greater than another or even that it is greater than a finite number, because if the infinite number were greater than, say, a million it would follow that on passing from the million to higher and higher numbers we would be approaching the infinite; but this is not so; on the contrary, the larger the number to which we pass, the more we recede from [this property of] infinity, because the greater the numbers the fewer [relatively] are the squares contained in them; but the squares in infinity cannot be less than the totality of all the numbers, as we have just agreed; hence the approach to greater and greater numbers means a departure from infinity.

I pass now to another consideration. Since lines and all continuous quantities are divisible into parts which are themselves divisible without end, I do not see how it is possible Edition: current; Page: [ 34 ] to avoid the conclusion that these lines are built up of an infinite number of indivisible quantities because a division and a subdivision which can be carried on indefinitely presupposes that the parts are infinite in number, otherwise the subdivision would reach an end; and if the parts are infinite in number, we must conclude that they are not finite in size, because an infinite number of finite quantities would give an infinite magnitude.

And thus we have a continuous quantity built up of an infinite number of indivisibles. But if we can carry on indefinitely the division into finite parts what necessity is there then for the introduction of non-finite parts?

The very fact that one is able to continue, without end, the division into finite parts [ in parti quante ] makes it necessary to regard the quantity as composed of an infinite number of immeasurably small elements [ di infiniti non quanti ]. Now in order to settle this matter I shall ask you to tell me whether, in your opinion, a continuum is made up of a finite or of an infinite number of finite parts [ parti quante ]. My answer is that their number is both infinite and finite; potentially infinite but actually finite [ infinite, in potenza; e finite, in atto ] ; that is to say, potentially infinite before division and actually finite after division; because parts cannot be said to exist in a body which is not yet divided or at least marked out; if this is not done we say that they exist potentially.

So that a line which is, for instance, twenty spans long is not said to contain actually twenty lines each one span in length except after division into twenty equal parts; before division it is said to contain them only potentially. Suppose the facts are as you say; tell me then whether, when the division is once made, the size of the original quantity is thereby increased, diminished, or unaffected. That is my opinion also. Therefore the finite parts [ parti quante ] in a continuum, whether actually or potentially present, do not make the quantity either larger or smaller; but it is perfectly clear that, if the number of finite parts actually Edition: current; Page: [ 35 ] contained in the whole is infinite in number, they will make the magnitude infinite.

Hence the number of finite parts, although existing only potentially, cannot be infinite unless the magnitude containing them be infinite; and conversely if the magnitude is Edition: NatlEd; Page: [ 81 ] finite it cannot contain an infinite number of finite parts either actually or potentially. How then is it possible to divide a continuum without limit into parts which are themselves always capable of subdivision? This distinction of yours between actual and potential appears to render easy by one method what would be impossible by another.

But I shall endeavor to reconcile these matters in another way; and as to the query whether the finite parts of a limited continuum [ continuo terminato ] are finite or infinite in number I will, contrary to the opinion of Simplicio, answer that they are neither finite nor infinite. This answer would never have occurred to me since I did not think that there existed any intermediate step between the finite and the infinite, so that the classification or distinction which assumes that a thing must be either finite or infinite is faulty and defective.

So it seems to me. And if we consider discrete quantities I think there is, between finite and infinite quantities, a third intermediate term which corresponds to every assigned number; so that if asked, as in the present case, whether the finite parts of a continuum are finite or infinite in number the best reply is that they are neither finite nor infinite but correspond to every assigned number.