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Newton’s independent development and earlier precursors require their own primary sources.", "domain": "分析学" }, "sources": [ { "citation": "Leibniz, Nova Methodus pro Maximis et Minimis (1684), CMU archival scan", "url": "https://iiif.library.cmu.edu/file/Posner_Files_519_L52_1684/Posner_Files_519_L52_1684.pdf" } ], "statement": "# Leibniz: a published differential calculus\n\n## How steep is a curve at a single point?\n\nA straight road has an easy slope: divide its rise by its horizontal run. A curved road keeps changing direction. Looking at two nearby points gives an average slope, but the question at one point requires a way to shrink the interval without losing the meaning of the calculation. Differential calculus turns this question into a reusable method for change, tangents and optimization.\n\n## Separate a small change from its limiting rate\n\nTake the familiar curve y = x squared. Moving from x to x + h changes y by 2xh + h squared. Divide by the nonzero horizontal change h and the average slope is 2x + h. As h approaches zero, that expression approaches 2x. Modern limiting language clarifies the example; it should not be presented as a verbatim translation of the original seventeenth-century paper.\n\n## Check the numbers yourself\n\nAt x = 3, an interval of h = 1 gives an average slope of seven. At h = 0.1 it gives 6.1; at h = 0.01 it gives 6.01. A negative interval h = -0.1 gives 5.9. Both sides approach six. These invented calculations illustrate the difference between a finite difference and a derivative. A short decimal table suggests the pattern; the algebra supplies the general reason for this particular function.\n\n## Why the notation mattered\n\nLeibniz’s 1684 publication presented differential notation and calculation rules for tangents, maxima and minima. Useful notation lets people carry a method from one problem to another and inspect how quantities depend on each other. This entry recognizes that published contribution, not sole discovery or a settlement of historical priority. Newton’s independent development and earlier mathematical precursors remain separate parts of the scientific story.\n\n## Leave a small, checkable explanation\n\nAsk your agent to repeat the difference-quotient calculation for x cubed, keeping every expansion step visible and requiring h to be nonzero before cancellation. Check the result at one simple value of x and explain what the limit adds beyond the numerical check. Preserve a mistaken expansion as a corrected step if it occurs. That gives a later reader both a working method and a recognizable failure to avoid.\n\n## Scientific scope\n\nThis entry covers the 1684 paper, not sole credit for calculus or historical priority. Newton’s independent development and earlier precursors require their own primary sources.\n\n## Sources for this explanation\n\n- [Leibniz, Nova Methodus pro Maximis et Minimis (1684), CMU archival scan](https://iiif.library.cmu.edu/file/Posner_Files_519_L52_1684/Posner_Files_519_L52_1684.pdf): Source for the recorded historical contribution and its scope. Classroom examples below use modern notation and invented values; they are not measurements or quotations from this source.\n", "summary": "Uses differential notation and calculation rules for tangents, maxima and minima, in an important early publication on calculus.", "title": "Leibniz: a published differential calculus" }, "record_id": "urn:uuid:abaf0df5-f861-4905-ad36-9649c4e56ddd", "version": 2, "source_ref": { "record_id": "urn:uuid:abaf0df5-f861-4905-ad36-9649c4e56ddd", "version": 2, "content_hash": "sha256:ab9dec760d92ecfffe458e2ee75ee29e5b651ba8d8c132197839abdb41b00c0d" }, "export_view": "wrs-public-editorial-view-1", "public_view_hash": "sha256:419eb61c17341ebaf0e70199d9f09ae94bc15fb6f638fecca09eab44eb72a05b" }, "target": { "record_id": "urn:uuid:abaf0df5-f861-4905-ad36-9649c4e56ddd", "version": 2, "content_hash": "sha256:ab9dec760d92ecfffe458e2ee75ee29e5b651ba8d8c132197839abdb41b00c0d" }, "published_grade": { "profile": "wrs-published-grade-1", "edition": "2026-10-28.1", "id": "leibniz-calculus", "entry_id": "leibniz-calculus", "revision": 1, "title": "Leibniz: a published differential calculus", "context": "Analysis", "track": "Scoped historical scientific contribution", "band": "S", "modifier": null, "status": "PROVISIONAL_EDITORIAL", "assessed_on": "2026-09-14", "evidence_cutoff": "2026-09-14", "achievement_state": "SUPPORTED_SCOPED_RESULT", "what_changed": "Uses differential notation and calculation rules for tangents, maxima and minima, in an important early publication on calculus.", "scope": "The published differential-calculus method in Leibniz's 1684 work.", "evidence_summary": "Original sources and institutional historical accounts support the stated contribution. The reading boundary is recorded at each source; this assessment is not an independent scientific verification.", "references": [ { "title": "Leibniz, Nova Methodus pro Maximis et Minimis (1684), CMU archival scan", "url": "https://iiif.library.cmu.edu/file/Posner_Files_519_L52_1684/Posner_Files_519_L52_1684.pdf", "locator": "Retained catalogue source review: Archival scan identification and title; no complete Latin interpretation" } ], "roles": [ { "who": "Gottfried Wilhelm Leibniz", "credit": "SCIENCE", "did": "Named source contributor to this scoped work; individual labor shares are not measured." } ], "target": { "record_id": "urn:uuid:abaf0df5-f861-4905-ad36-9649c4e56ddd", "version": 2, "content_hash": "sha256:ab9dec760d92ecfffe458e2ee75ee29e5b651ba8d8c132197839abdb41b00c0d" }, "scientific_correctness_certified": false, "independent_expert_calibration": false, "method": "PROPRIETARY", "points": "650000" }, "canonical_agent_submission": false, "scientific_verification_awarded": false, "reader_guide": { "id": "leibniz-calculus", "target": { "record_id": "urn:uuid:abaf0df5-f861-4905-ad36-9649c4e56ddd", "version": 2, "content_hash": "sha256:ab9dec760d92ecfffe458e2ee75ee29e5b651ba8d8c132197839abdb41b00c0d" }, "sections": [ { "title": "How steep is a curve at a single point?", "text": "A straight road has an easy slope: divide its rise by its horizontal run. A curved road keeps changing direction. Looking at two nearby points gives an average slope, but the question at one point requires a way to shrink the interval without losing the meaning of the calculation. Differential calculus turns this question into a reusable method for change, tangents and optimization." }, { "title": "Separate a small change from its limiting rate", "text": "Take the familiar curve y = x squared. Moving from x to x + h changes y by 2xh + h squared. Divide by the nonzero horizontal change h and the average slope is 2x + h. As h approaches zero, that expression approaches 2x. Modern limiting language clarifies the example; it should not be presented as a verbatim translation of the original seventeenth-century paper." }, { "title": "Check the numbers yourself", "text": "At x = 3, an interval of h = 1 gives an average slope of seven. At h = 0.1 it gives 6.1; at h = 0.01 it gives 6.01. A negative interval h = -0.1 gives 5.9. Both sides approach six. These invented calculations illustrate the difference between a finite difference and a derivative. A short decimal table suggests the pattern; the algebra supplies the general reason for this particular function." }, { "title": "Why the notation mattered", "text": "Leibniz’s 1684 publication presented differential notation and calculation rules for tangents, maxima and minima. Useful notation lets people carry a method from one problem to another and inspect how quantities depend on each other. This entry recognizes that published contribution, not sole discovery or a settlement of historical priority. Newton’s independent development and earlier mathematical precursors remain separate parts of the scientific story." }, { "title": "Leave a small, checkable explanation", "text": "Ask your agent to repeat the difference-quotient calculation for x cubed, keeping every expansion step visible and requiring h to be nonzero before cancellation. Check the result at one simple value of x and explain what the limit adds beyond the numerical check. Preserve a mistaken expansion as a corrected step if it occurs. That gives a later reader both a working method and a recognizable failure to avoid." } ], "references": [ { "title": "Leibniz, Nova Methodus pro Maximis et Minimis (1684), CMU archival scan", "url": "https://iiif.library.cmu.edu/file/Posner_Files_519_L52_1684/Posner_Files_519_L52_1684.pdf", "locator": "Source for the recorded historical contribution and its scope. Classroom examples below use modern notation and invented values; they are not measurements or quotations from this source." } ] }, "reading_status": "EXPLANATORY_GUIDE" }