The Online Library of Liberty

A project of Liberty Fund, Inc.

• Herbert Spencer’s Works.
• Note By the Trustees.
• Preface.
• Note.
• Family-antecedents.
• I.: Extraction.
• II.: Grandparents and Their Children.
• III.: Parents.
• Part I.: 1820—1837.
• Chapter I.: Childhood. 1820—1827. Æt. 1—7.
• Chapter II.: Boyhood. 1827—33. Æt. 7—13.
• Chapter III.: A Journey and a Flight. 1833. Æt. 13.
• Chapter IV.: Youth At Hinton. 1833—36. Æt. 13—16.
• Chapter V.: A False Start. 1836—37. Æt. 16—17.
• Part II.: 1837—1841
• Chapter VI.: Commence Engineering. 1837—38. Æt. 17—18.
• Chapter VII.: Life At Worcester. 1838—40. Æt. 18—20.
• Chapter VIII.: Some Months At Powick. 1840. Æt. 20.
• Chapter IX.: A Nomadic Period. 1840—41. Æt. 20—21.
• Part III.: 1841—1844
• Chapter X.: Return to Derby. 1841—42. Æt. 21—2.
• Chapter XI.: A Visit and Its Consequences. 1842. Æt. 22.
• Chapter XII.: Back At Home. 1842—43. Æt. 22—23.
• Chapter XIII.: A Campaign In London. 1843. Æt. 23.
• Chapter XIV.: At Home Again. 1843—44. Æt. 23—24.
• Part IV.: 1844—1848
• Chapter XV.: A Brief Sub-editorship. 1844. Æt. 24.
• Chapter XVI.: A Parliamentary Survey. 1844. Æt. 24.
• Chapter XVII.: An Interval In Town. 1845. Æt. 24—5.
• Chapter XVIII.: Another Parliamentary Survey. 1845—46. Æt. 25—26.
• Chapter XIX.: Inventions. 1846—47. Æt. 26—27.
• Chapter Xix A.: Suspense. 1848. Æt. 28.
• Part V.: 1848—1853
• Chapter XX.: Begin Journalism. 1848—50. Æt. 28—30.
• Chapter XXI.: My First Book. 1848—50 Æt. 28—30.
• Chapter XXII.: An Idle Year. 1850—51. Æt. 30—31.
• Chapter XXIII.: A More Active Year. 1852. Æt. 31—32.
• Chapter XXIV.: Leave the Economist. 1853. Æt. 33—34.
• Part VI.: 1853—1856
• Chapter XXV.: Two Months’ Holiday. 1853. Æt. 33.
• Chapter XXVI.: Writing For Quarterly Reviews. 1853—54. Æt. 33—34.
• Chapter XXVII.: My Second Book. 1854—55. Æt. 34—35.
• Chapter XXVIII.: Eighteen Months Lost. 1855—56. Æt. 35—36.
• Chapter XXIX.: Some Significant Essays. 1856—7. Æt. 36—37.
• Appendices.
• Appendix A.: Skew Arches.
• Appendix B.: Geometrical Theorem.
• Appendix C.: Velocimeter. an Instrument For Calculating Velocities On Railways, &c.
• Appendix D.: Scale of Equivalents.
• Appendix E.: Ideas About a Universal Language.
• Appendix F.: Remarks On the Theory of Reciprocal Dependence In the Animal and Vegetable Creations As Regards Its Bearing Upon PalÆontology.
• Appendix G.: Levelling Appliances.
• Appendix H.: On a Proposed Cephalograph.
• Appendix I.: Registered Binding-pins, For Securing Music and Unbound Publications.
• Appendix J.: the Form of the Earth No Proof of Original Fluidity.
• Appendix K.: Letter to the Athen Æum Concerning the Misstatements of the Rev. T. Mozley.

APPENDIX B.

GEOMETRICAL THEOREM.

[From the “Civil Engineer and Architect’s Journal” for July, 1840.]

Sir,—

I believe that the following curious property of a circle has not hitherto been noticed; or if it has, I am not aware of its existence in any of our works on Geometry.

Let ABCDE be a circle, of which ACD is any given segment: Let any number of triangles ABD, ACD, &c., be drawn in this segment, and let circles be inscribed in these triangles: their centres F, G, &c. are in the arc of a circle, whose centre is at E, the middle of the arc of the opposite segment AED.

Demonstration.

Join AF, FD, AG, GD; then since F is the centre of the circle, inscribed in the triangle ABD, the lines AF, FD, bisect the angles BAD, BDA. (Euc. B. 4, P. 4). For a like reason AG, GD, bisect the angles CAD, CDA; hence the angles FAD, FDA, together, are equal to half the angles, BAD, BDA together, and the angles GAD, GDA together, to half the angles CAD, CDA together. Now the angles ABD, ACD, are equal (being in the same segment) therefore the angles BAD, BDA together, are equal to the angles CAD, CDA together, and as the halves of equals are equal, the angles FAD, FDA together are equal to the angles GAD, GDA together; that is in the two triangles AFD, AGD, two angles of the one, are together equal to two angles of the other, and therefore the third angle AFD, is equal to the third angle AGD. The same reasoning will prove that all angles similarly circumstanced to AFD, are also equal to AGD: therefore, the points A, F, G, D, are in an arc of a circle.

Join BF, and produce it to cut the opposite circumference in E and join EA, ED; then because the angle ABE, is equal to the angle DBE, the segment AE, is equal to the segment ED, and the chord AE, to the chord ED. Again the angles ABE, EDA, are equal (being in the same segment), and by construction, the angle ADF is equal to the angle FDB, therefore the whole angle EDF, is equal to the two ABF, FDB, that is to the two FBD, FDB, that is to the exterior angle EFD; therefore the angle EFD, is equal to the angle EDF; consequently EF, is equal to ED, that is to EA. The same reasoning would prove EF to be equal to a line drawn from G, to the point E. Wherefore the point E is the centre of a circle, of which F and G, as also the centres of all other circles similarly inscribed, are in the circumference.

H. Spencer.

• Birmingham and Gloucester Railway Office, Worcester.