For instance, although I cannot find the source-I do believe the Vilna Gaon told a Talmid of his to translate the Elements of Euclid (Geometric proofs, inter Alia) to Hebrew in order to augment understanding in sugyot found in succah, eruvin, etc. It is called “״איל משלש.
Are there works similarly extracted from the non-Jews and reproduced or incorporated into a corpus for the purposes of aiding Limud Torah?
This is not a specific work, but an example of using secular knowledge for Torah:
והאמר רב שמונה עשר חדשים גדלתי אצל רועה בהמה לידע איזה מום קבוע ואיזה מום עובר
But did not Rab himself say: I spent eighteen months with a shepherd in order to learn which was a permanent and which a passing blemish?
The gemara in Shabbos 85a tells us that Chazzal had a tradition going back to biblical times of knowledge they learned from various nations concerning plant growth, namely how far they need plant various species to avoid Kilaim.
וקים להו לרבנן דחמשא בשיתא לא ינקי מהדדי
And the Sages have an accepted tradition that five seeds in a six-handbreadth space do not draw sustenance from one another.
ומנלן דהא דקים להו לרבנן (דחמשא בשיתא) מילתא היא דאמר רבי חייא בר אבא אמר רבי יוחנן מאי דכתיב לא תסיג גבול רעך [אשר גבלו ראשונים] גבול שגבלו ראשונים לא תסיג מאי גבלו ראשונים אמר רבי שמואל בר נחמני אמר רבי יונתן (מאי) דכתיב אלה בני שעיר החרי ישבי הארץ אטו כולי עלמא יושבי רקיע נינהו אלא שהיו בקיאין בישובה של ארץ שהיו אומרים מלא קנה זה לזית מלא קנה זה לגפנים מלא קנה זה לתאנים וחרי שמריחים את הארץ וחוי אמר רב פפא שהיו טועמין את הארץ כחויא רב אחא בר יעקב אמר חרי שנעשו בני חורין מנכסיהן The Gemara asks: And from where do we derive that when the Sages have an accepted tradition it is a substantial matter, meaning that the tradition is reliable? Rabbi Ḥiyya bar Abba said that Rabbi Yoḥanan said: What is the meaning of that which is written: “You shall not cross your neighbor’s border, which they of the old times have set in your inheritance that you shall inherit” (Deuteronomy 19:14)? It means that you shall not cross the border that the early generations set, establishing the parameters necessary for each plant. The Gemara asks: What is the meaning of the phrase: The early generations set? Rabbi Shmuel bar Naḥmani said that Rabbi Yonatan said: What is the meaning of that which was written: “These are the sons of Seir the Horite who inhabit the land, Lotan and Shoval and Zibeon and Ana” (Genesis 36:20)? And is everyone else inhabitants of the heavens, that it was necessary for the verse to emphasize that these inhabit the land? Rather, it means that they were experts in the settlement of the land, as they would say: This tract of land that is the full length of a rod is fit for olive trees; this full length of a rod is fit for grapes, this full length of a rod is fit for figs. And the members of this tribe were called Horites [ḥori] since they smelled [heriḥu] the earth to determine what is fit to be grown there. The allusion is based on a transposition of the letters ḥet and reish.
Rambam in chapter 17 halacha 24 of Kidush Hachodesh tells us that all the calculations he quoted to explain kidush hachodesh are from the Greeks as we lost our calculations from Jewish sources.
In 1881 Reb Yisroel Salanter used the recently discovered idea of the subconscious to explain some words of Chazal. You can read a translation of his letter starting on page 35 of this PDF.
He sent this letter as a way to raise support for his yeshiva.
The first half of that PDF is also somewhat related to this topic. It was a letter by Rav Hirsch concerning Agadata. Some points he made were that chazzal themselves at time in Agadata were merely quoting scientific ideas of their times and adapting them self improvement.
Rambam, Hilchos Kiddush Hachodesh 17:24 (after spending several chapters on the details of astronomical calculations):
וטעם כל אלו החשבונות ומפני מה מוסיפים מנין זה ומפני מה גורעין. והיאך נודע כל דבר ודבר מאלו הדברים. והראיה על כל דבר ודבר. היא חכמת התקופות והגימטריות שחברו בה חכמי יון ספרים הרבה והם הנמצאים עכשיו ביד החכמים... שכל דבר שנתגלה טעמו ונודעה אמיתתו בראיות שאין בהם דופי אנו סומכין על זה האיש שאמרו או שלמדו על הראיה שנתגלתה והטעם שנודע
The rationales for all these calculations, and the reasons why this number is added, and why that subtraction is made, and how all these concepts are known, and the proofs for each of these principles are [the subject] of the wisdom of astronomy and geometry, concerning which the Greeks wrote many books. These texts are presently in the hands of the sages... For a matter whose rationale has been revealed and has proven truthful in an unshakable manner, we do not rely on [the personal authority of] the individual who made these statements or taught these concepts, but on the proofs he presented and the reasons he made known.
The Ramchal writes in his introduction to Sefer HaHigaion (the Book of Logic):
ובראותי הצורך הגדול אשר לנו באמת בדבר הזה, שבלעדיו אי אפשר לכנס אל חדרי החכמות ולהשתשע בתענוגיהם כראוי, בחרתי לסדר זה הלימוד בקיצור כפי שראיתי מוכרח לתשלום הענין, רובו העתקתיו מספרים שקדמוני בלשונות אחרות.
And when I saw the great need that we truly have for [formal logic], for without it it is impossible to enter the chambers of wisdom and to rightfully delight in their pleasures,I decided to arrange this study concisely as I saw necessary to cover the topic; mostly copied from works that came before me in other languages.
The work is based on Aristotle.
Bereishis Rabbah 14:2:
בָּעוֹן קַמֵּיהּ דְּרַבִּי אַבָּהוּ, מִנַּיִן שֶׁהַנּוֹצָר לְשִׁבְעָה חַי, אֲמַר לְהוֹן מִדִּידְכוֹן אֲנָא מַמְטֵי לְכוֹן, זיט"א אפט"א, איט"א אוכט"א.
[Some people] asked R’ Avahu: “From where do you know that a fetus born at seven months can live?” He said to them, “From your own [language] I will respond to you: zeta [the seventh letter of the Greek alphabet] means “life,” and eta [the eighth letter of the Greek alphabet] means “death.”
The Chumash describes how the Jews used secular poetry to determine which lands previously owned by Moav they were allowed to conquer.
(I'm assuming that when you said Tanaim, Amoraim, Rishonim, Achronim, it was lav davka and you weren't intending to exclude earlier sources.)
Megilat Aicha tells us: "If someone tells you, 'There is wisdom among the Goyiim, you can beleive that... "
This is copied off the internet: In the Book of Lamentations, the prophet describes the situation of the people of Israel after the destruction of the Temple:" ... A queen and her ministers are among the nations, there is no Torah, Even its prophets did not find a vision from God "(Chapter 2, verse 9) The people of Israel after the destruction was under foreign rule, without a leader from within it, and as a result there is no Torah teacher (Rashi) "If a person says to you: There is wisdom in the nations - believe ... (And if a man says to you," There is Torah among the nations - do not believe, As it is written: A queen and her ministers in the nations have no Torah "(Med In the Diaspora, when this connection was severed, the prophecy ceased, and to a certain extent the light of the Torah was also extinguished. "The Torah is given to the people of Israel only and belongs only to him, and the people of Israel are the link between humanity and the Creator of the world. The Torah with human intelligence - is a prophetic level that belongs only to the Jewish people. Under foreign rule, the Torah is lacking. On the other hand, human intellectual wisdom and life experience are things that every person can have from every nation and language. It is told in the Gemara that Rabbi Yochanan used to rise from the elders of the gentiles because of the vast experience of life that they accumulated throughout their lives (Tractate Kiddushin 35a at the end.) The Rambam at the beginning of his introduction to tractate Avot (8 chapters) writes: The things that I will say in these chapters ... are matters that are compiled from the words of the Sages (sages of Israel) ... and from the philosophers, as well as from the writings of many people. On the Greek philosopher Aristotle, Maimonides writes: "He who taught men the ways of proof and its laws and its conditions" ("Guide of the Perplexed" Part II Chapter 15). The Torah belongs only to us, to the people of Israel. Human wisdom, as it is, belongs to the whole human race, and it is worthwhile to learn it from anyone who has something of it. Sincerely, Uri, The staff of Yeshivat Har Etzion
