...concerning the Geometric discovery of two mean proportionals long since made, and concerning the truth and certainty of all stereometric problems proceeding from the geometrically demonstrated discovery of two proportionals. Further matters may be seen in the cited location.

I add that the ancients also had the method and use of constructing a Caliber Rule, as is evident from the letter of Eratosthenes to King Ptolemy, which Bettinus cites in the same place mentioned above. For thus it is there:

"But we have devised, through instruments, an easy discovery by which we might find not only two mean proportionals for two given ones, but as many as might be proposed; and that having been found, we will finally be able to reduce a proposed solid contained by equidistant lines to a cube, or even form another figure from one that is either equal or greater while maintaining similarity. Since it is doubtful to no one that by an instrument of this kind altars and buildings can be doubled, and the measures of liquids and dry goods, such as bushels and the like, can be referred to a cube, by the sides of whose vessels the capacity of the measures is known; and to speak summarily, the knowledge of this question is useful to those wishing to double or make greater the instruments from which cloths, stones, or iron balls are thrown. For it is necessary for all things to grow in breadth and in length by a certain proportion, whether they be holes, or ropes, and other inserted things, or whatever is required, if we desire the whole to be increased in proportion, which cannot be done without the discovery of the mean."

CHAPTER III.

From such a numerous crowd of Pyrotechnicians in this age of ours, you would not find even one (I dare say) who is not a practitioner, devoid of much knowledge, and although very experienced in his own art (which he did not learn at home during time of peace, at leisure, among the desired delights of body and mind; but in the field, through the greatest dangers of life, and in clearly bloody sweat), does not wish not so much to be, as to seem and be held by others to be, something. Indeed, I have been able to know many who did not consider themselves worthy of being called simple and common, but rather "Field Pyrotechnicians" (commonly Feld-Feuerwercker Field Pyrotechnicians). Hence, totally renouncing the rudiments of theory and divine Mathematics, they deem it the greatest disgrace if anyone dedicated to Pyrotechnics brings forward Archimedean or Euclidean theorems and demonstrations, or those of other most excellent Geometers, to test and stabilize the rules of the art. Hence that new science, unknown for many centuries past, has been born: Pseudo-mechanics, whose chief and general axiom is: To do everything perturbedly, confusedly, and nothing to the point. But the sweetest offspring of such a mother is: Daily and inexplicable errors (both in constructing war machines and handling them correctly, as well as in preparing artificial fires, celebrations, and games), joined with the greatest loss to Princes and the danger to the lives of both the artists themselves and the spectators. But how miserable those are who are ignorant of true Mathematics and its elements, let it be pleasing to hear Paul Guldin speaking in his Centrobaricorum, Book 4, Chapter 5, concerning an Arithmetical problem. For thus he says:

"Lest, therefore, our Philomathematicians be rendered unworthy of this name, but rather emerge from the ocean of ignorance and be kindled to the study of those most noble sciences, we have observed Mathematics as a most powerful Queen with a numerous retinue of subject sciences brought forth at the beginning of our lectures..."