Essays On Education And Kindred Subjects (Fiscle Part- 11) by Herbert Spencer (best fiction novels to read TXT) 📕

Between The   Sides And Angles Of    Triangles--_Trigonometry_ A Subdivision

Of Pure Mathematics. Further, The   Reduction Of    The   Doctrine Of    The

Sphere To The   Quantitative Form Needed For Astronomical Purposes,

Required The   Formation Of    A _Spherical Trigonometry_, Which Was Also

Achieved By Hipparchus. Thus Both Plane And Spherical Trigonometry,

Which Are Parts Of    The   Highly Abstract And Simple Science Of    Extension,

Remained Undeveloped Until The   Less Abstract And More Complex Science Of

The Celestial Motions Had Need Of    Them. The   Fact Admitted By M. Comte,

That Since Descartes The   Progress Of    The   Abstract Division Of

Mathematics Has Been Determined By That Of    The   Concrete Division, Is

Paralleled By The   Still More Significant Fact That Even Thus Early The

Progress Of    Mathematics Was Determined By That Of    Astronomy.

And Here, Indeed, We May See Exemplified The   Truth, Which The   Subsequent

History Of    Science Frequently Illustrates, That Before Any More

Abstract Division Makes A Further Advance, Some More Concrete Division

Must Suggest The   Necessity For That Advance--Must Present The   New Order

Of Questions To Be Solved. Before Astronomy Presented Hipparchus With

The Problem Of    Solar Tables, There Was Nothing To Raise The   Question Of

The Relations Between Lines And Angles; The   Subject-Matter Of

Trigonometry Had Not Been Conceived. And As There Must Be Subject-Matter

Before There Can Be Investigation, It Follows That The   Progress Of    The

Concrete Divisions Is As Necessary To That Of    The   Abstract, As The

Progress Of    The   Abstract To That Of    The   Concrete.

Just Incidentally Noticing The   Circumstance That The   Epoch We Are

Describing Witnessed The   Evolution Of    Algebra, A Comparatively Abstract

Division Of    Mathematics, By The   Union Of    Its Less Abstract Divisions,

Geometry And Arithmetic--A Fact Proved By The   Earliest Extant Samples Of

Algebra, Which Are Half Algebraic, Half Geometric--We Go On To Observe

That During The   Era In Which Mathematics And Astronomy Were Thus

Advancing, Rational Mechanics Made Its Second Step; And Something Was

Done Towards Giving A Quantitative Form To Hydrostatics, Optics, And

Harmonics. In Each Case We Shall See, As Before, How The   Idea Of

Equality Underlies All Quantitative Prevision; And In What Simple Forms

This Idea Is First Applied.

As Already Shown, The   First Theorem Established In Mechanics Was, That

Equal Weights Suspended From A Lever With Equal Arms Would Remain In

Equilibrium. Archimedes Discovered That A Lever With Unequal Arms Was In

Equilibrium When One Weight Was To Its Arm As The   Other Arm To Its

Weight; That Is--When The   Numerical Relation Between One Weight And Its

Arm Was _Equal_ To The   Numerical Relation Between The   Other Arm And Its

Weight.

The First Advance Made In Hydrostatics, Which We Also Owe To Archimedes,

Was The   Discovery That Fluids Press _Equally_ In All Directions; And

From This Followed The   Solution Of    The   Problem Of    Floating Bodies:

Namely, That They Are In Equilibrium When The   Upward And Downward

Pressures Are _Equal_.

In Optics, Again, The   Greeks Found That The   Angle Of    Incidence Is

_Equal_ To The   Angle Of    Reflection; And Their Knowledge Reached No

Further Than To Such Simple Deductions From This As Their Geometry

Sufficed For. In Harmonics They Ascertained The   Fact That Three Strings

Of _Equal_ Lengths Would Yield The   Octave, Fifth And Fourth, When

Strained By Weights Having Certain Definite Ratios; And They Did Not

Progress Much Beyond This. In The   One Of    Which Cases We See Geometry

Used In Elucidation Of    The   Laws Of    Light; And In The   Other, Geometry And

Arithmetic Made To Measure The   Phenomena Of    Sound.

Did Space Permit, It Would Be Desirable Here To Describe The   State Of

The Less Advanced Sciences--To Point Out How, While A Few Had Thus

Reached The   First Stages Of    Quantitative Prevision, The   Rest Were

Progressing In Qualitative Prevision--How Some Small Generalisations

Were Made Respecting Evaporation, And Heat, And Electricity, And

Magnetism, Which, Empirical As They Were, Did Not In That Respect Differ

From The   First Generalisations Of    Every Science--How The   Greek

Physicians Had Made Advances In Physiology And Pathology, Which,

Considering The   Great Imperfection Of    Our Present Knowledge, Are By No

Means To Be Despised--How Zoology Had Been So Far Systematised By

Aristotle, As, To Some Extent, Enabled Him From The   Presence Of    Certain

Organs To Predict The   Presence Of    Others--How In Aristotle's _Politics_

There Is Some Progress Towards A Scientific Conception Of    Social

Phenomena, And Sundry Previsions Respecting Them--And How In The   State

Of The   Greek Societies, As Well As In The   Writings Of    Greek

Philosophers, We May Recognise Not Only An Increasing Clearness In That

Conception Of    Equity On Which The   Social Science Is Based, But Also Some

Appreciation Of    The   Fact That Social Stability Depends Upon The

Maintenance Of    Equitable Regulations. We Might Dwell At Length Upon The

Causes Which Retarded The   Development Of    Some Of    The   Sciences, As, For

Example, Chemistry; Showing That Relative Complexity Had Nothing To Do

With It--That The   Oxidation Of    A Piece Of    Iron Is A Simpler Phenomenon

Than The   Recurrence Of    Eclipses, And The   Discovery Of    Carbonic Acid Less

Difficult Than That Of    The   Precession Of    The   Equinoxes--But That The

Relatively Slow Advance Of    Chemical Knowledge Was Due, Partly To The

Fact That Its Phenomena Were Not Daily Thrust On Men's Notice As Those

Of Astronomy Were; Partly To The   Fact That Nature Does Not Habitually

Supply The   Means, And Suggest The   Modes Of    Investigation, As In The

Sciences Dealing With Time, Extension, And Force; And Partly To The   Fact

That The   Great Majority Of    The   Materials With Which Chemistry Deals,

Instead Of    Being Ready To Hand, Are Made Known Only By The   Arts In Their

Slow Growth; And Partly To The   Fact That Even When Known, Their Chemical

Properties Are Not Self-Exhibited, But Have To Be Sought Out By

Experiment.

Merely Indicating All These Considerations, However, Let Us Go On To

Contemplate The   Progress And Mutual Influence Of    The   Sciences In Modern

Days; Only Parenthetically Noticing How, On The   Revival Of    The

Scientific Spirit, The   Successive Stages Achieved Exhibit The   Dominance

Of The   Same Law Hitherto Traced--How The   Primary Idea In Dynamics, A

Uniform Force, Was Defined By Galileo To Be A Force Which Generates

_Equal_ Velocities In _Equal_ Successive Times--How The   Uniform Action

Of Gravity Was First Experimentally Determined By Showing That The   Time

Elapsing Before A Body Thrown Up, Stopped, Was _Equal_ To The   Time It

Took To Fall--How The   First Fact In Compound Motion Which Galileo

Ascertained Was, That A Body Projected Horizontally Will Have A Uniform

Motion Onwards And A Uniformly Accelerated Motion Downwards; That Is,

Will Describe _Equal_ Horizontal Spaces In _Equal_ Times, Compounded

With _Equal_ Vertical Increments In _Equal_ Times--How His Discovery

Respecting The   Pendulum Was, That Its Oscillations Occupy _Equal_

Intervals Of    Time Whatever Their Length--How The   Principle Of    Virtual

Velocities Which He Established Is, That In Any Machine The   Weights That

Balance Each Other Are Reciprocally As Their Virtual Velocities; That

Is, The   Relation Of    One Set Of    Weights To Their Velocities _Equals_ The

Relation Of    The   Other Set Of    Velocities To Their Weights; And How Thus

His Achievements Consisted In Showing The   Equalities Of    Certain

Magnitudes And Relations, Whose Equalities Had Not Been Previously

Recognised.

When Mechanics Had Reached The   Point To Which Galileo Brought It--When

The Simple Laws Of    Force Had Been Disentangled From The   Friction And

Atmospheric Resistance By Which All Their Earthly Manifestations Are

Disguised--When Progressing Knowledge Of    _Physics_ Had Given A Due

Insight Into These Disturbing Causes--When, By An Effort Of    Abstraction,

It Was Perceived That All Motion Would Be Uniform And Rectilinear Unless

Interfered With By External Forces--And When The   Various Consequences Of

This Perception Had Been Worked Out; Then It Became Possible, By The

Union Of    Geometry And Mechanics, To Initiate Physical Astronomy.

Geometry And Mechanics Having Diverged From A Common Root In Men's

Sensible Experiences; Having, With Occasional Inosculations, Been

Separately Developed, The   One Partly In Connection With Astronomy, The

Other Solely By Analysing Terrestrial Movements; Now Join In The

Investigations Of    Newton To Create A True Theory Of    The   Celestial

Motions. And Here, Also, We Have To Notice The   Important Fact That, In

The Very Process Of    Being Brought Jointly To Bear Upon Astronomical

Problems, They Are Themselves Raised To A Higher Phase Of    Development.

For It Was In Dealing With The   Questions Raised By Celestial Dynamics

That The   Then Incipient Infinitesimal Calculus Was Unfolded By Newton

And His Continental Successors; And It Was From Inquiries Into The

Mechanics Of    The   Solar System That The   General Theorems Of    Mechanics

Contained In The   _Principia_,--Many Of    Them Of    Purely Terrestrial

Application--Took Their Rise. Thus, As In The   Case Of    Hipparchus, The

Presentation Of    A New Order Of    Concrete Facts To Be Analysed, Led To The

Discovery Of    New Abstract Facts; And These Abstract Facts Having Been

Laid Hold Of, Gave Means Of    Access To Endless Groups Of    Concrete Facts

Before Incapable Of    Quantitative Treatment.

Meanwhile, Physics Had Been Carrying Further That Progress Without

Which, As Just Shown, Rational Mechanics Could Not Be Disentangled. In

Hydrostatics, Stevinus Had Extended And Applied The   Discovery Of

Archimedes. Torricelli Had Proved Atmospheric Pressure, "By Showing That

This Pressure Sustained Different Liquids At Heights Inversely

Proportional To Their Densities;" And Pascal "Established The   Necessary

Diminution Of    This Pressure At Increasing Heights In The   Atmosphere:"

Discoveries Which In Part Reduced This Branch Of    Science To A

Quantitative Form. Something Had Been Done By Daniel Bernouilli Towards

The Dynamics Of    Fluids. The   Thermometer Had Been Invented; And A Number

Of Small Generalisations Reached By It. Huyghens And Newton Had Made

Considerable Progress In Optics; Newton Had Approximately Calculated The

Rate Of    Transmission Of    Sound; And The   Continental Mathematicians Had

Succeeded In Determining Some Of    The   Laws Of    Sonorous Vibrations.

Magnetism And Electricity Had Been Considerably Advanced By Gilbert.

Chemistry Had Got As Far As The   Mutual Neutralisation Of    Acids And

Alkalies. And Leonardo Da Vinci Had Advanced In Geology To The

Conception Of    The   Deposition Of    Marine Strata As The   Origin Of    Fossils.

Our Present Purpose Does Not Require That We Should Give Particulars.

All That It Here Concerns Us To Do Is To Illustrate The   _Consensus_

Subsisting In This Stage Of    Growth, And Afterwards. Let Us Look At A Few

Cases.

The Theoretic Law Of    The   Velocity Of    Sound Enunciated By Newton On

Purely Mechanical Considerations, Was Found Wrong By One-Sixth. The

Error Remained Unaccounted For Until The   Time Of    Laplace, Who,

Suspecting That The   Heat Disengaged By The   Compression Of    The   Undulating

Strata Of    The   Air, Gave Additional Elasticity, And So Produced The

Difference, Made The   Needful Calculations And Found He Was Right. Thus

Acoustics Was Arrested Until Thermology Overtook And Aided It. When

Boyle And Marriot Had Discovered The   Relation Between The   Density Of

Gases And The   Pressures They Are Subject To; And When It Thus Became

Possible To Calculate The   Rate Of    Decreasing Density In The   Upper Parts

Of The   Atmosphere, It Also Became Possible To Make Approximate Tables Of

The Atmospheric Refraction Of    Light. Thus Optics, And With It Astronomy,

Advanced With Barology. After The   Discovery Of    Atmospheric Pressure Had

Led To The   Invention Of    The   Air-Pump By Otto Guericke; And After It Had

Become Known That Evaporation Increases In Rapidity As Atmospheric

Pressure Decreases; It Became Possible For Leslie, By Evaporation In A

Vacuum, To Produce The   Greatest Cold Known; And So To Extend Our

Knowledge Of    Thermology By Showing That There Is No Zero Within Reach Of

Our Researches. When Fourier Had Determined The   Laws Of    Conduction Of

Heat, And When The