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

Beyond Ten--The Limit Of    The   Simple Finger Notation. The   Fact That In So

Many Instances, Remote, And Seemingly Unrelated Nations, Have Adopted

_Ten_ As Their Basic Number; Together With The   Fact That In The

Remaining Instances The   Basic Number Is Either _Five_ (The Fingers Of

One Hand) Or _Twenty_ (The Fingers And Toes); Almost Of    Themselves Show

That The   Fingers Were The   Original Units Of    Numeration. The   Still

Surviving Use Of    The   Word _Digit_, As The   General Name For A Figure In

Arithmetic, Is Significant; And It Is Even Said That Our Word _Ten_

(Sax. _Tyn_; Dutch, _Tien_; German, _Zehn_) Means In Its Primitive

Expanded Form _Two Hands_. So That Originally, To Say There Were Ten

Things, Was To Say There Were Two Hands Of    Them.

From All Which Evidence It Is Tolerably Clear That The   Earliest Mode Of

Conveying The   Idea Of    Any Number Of    Things, Was By Holding Up As Many

Fingers As There Were Things; That Is--Using A Symbol Which Was _Equal_,

In Respect Of    Multiplicity, To The   Group Symbolised. For Which Inference

There Is, Indeed, Strong Confirmation In The   Recent Statement That Our

Own Soldiers Are Even Now Spontaneously Adopting This Device In Their

Dealings With The   Turks. And Here It Should Be Remarked That In This

Recombination Of    The   Notion Of    Equality With That Of    Multiplicity, By

Which The   First Steps In Numeration Are Effected, We May See One Of    The

Earliest Of    Those Inosculations Between The   Diverging Branches Of

Science, Which Are Afterwards Of    Perpetual Occurrence.

Indeed, As This Observation Suggests, It Will Be Well, Before Tracing

The Mode In Which Exact Science Finally Emerges From The   Merely

Approximate Judgments Of    The   Senses, And Showing The   Non-Serial

Evolution Of    Its Divisions, To Note The   Non-Serial Character Of    Those

Preliminary Processes Of    Which All After Development Is A Continuation.

On Reconsidering Them It Will Be Seen That Not Only Are They Divergent

Growths From A Common Root, Not Only Are They Simultaneous In Their

Progress; But That They Are Mutual Aids; And That None Can Advance

Without The   Rest. That Completeness Of    Classification For Which The

Unfolding Of    The   Perceptions Paves The   Way, Is Impossible Without A

Corresponding Progress In Language, By Which Greater Varieties Of

Objects Are Thinkable And Expressible. On The   One Hand It Is Impossible

To Carry Classification Far Without Names By Which To Designate The

Classes; And On The   Other Hand It Is Impossible To Make Language Faster

Than Things Are Classified.

Again, The   Multiplication Of    Classes And The   Consequent Narrowing Of

Each Class, Itself Involves A Greater Likeness Among The   Things Classed

Together; And The   Consequent Approach Towards The   Notion Of    Complete

Likeness Itself Allows Classification To Be Carried Higher. Moreover,

Classification Necessarily Advances _Pari Passu_ With Rationality--The

Classification Of    _Things_ With The   Classification Of    _Relations_. For

Things That Belong To The   Same Class Are, By Implication, Things Of

Which The   Properties And Modes Of    Behaviour--The Co-Existences And

Sequences--Are More Or Less The   Same; And The   Recognition Of    This

Sameness Of    Co-Existences And Sequences Is Reasoning. Whence It Follows

That The   Advance Of    Classification Is Necessarily Proportionate To The

Advance Of    Generalisations. Yet Further, The   Notion Of    _Likeness_, Both

In Things And Relations, Simultaneously Evolves By One Process Of

Culture The   Ideas Of    _Equality_ Of    Things And _Equality_ Of    Relations;

Which Are The   Respective Bases Of    Exact Concrete Reasoning And Exact

Abstract Reasoning--Mathematics And Logic. And Once More, This Idea Of

Equality, In The   Very Process Of    Being Formed, Necessarily Gives Origin

To Two Series Of    Relations--Those Of    Magnitude And Those Of    Number: From

Which Arise Geometry And The   Calculus. Thus The   Process Throughout Is

One Of    Perpetual Subdivision And Perpetual Intercommunication Of    The

Divisions. From The   Very First There Has Been That _Consensus_ Of

Different Kinds Of    Knowledge, Answering To The   _Consensus_ Of    The

Intellectual Faculties, Which, As Already Said, Must Exist Among The

Sciences.

Let Us Now Go On To Observe How, Out Of    The   Notions Of    _Equality_ And

_Number_, As Arrived At In The   Manner Described, There Gradually Arose

The Elements Of    Quantitative Prevision.

Equality, Once Having Come To Be Definitely Conceived, Was Readily

Applicable To Other Phenomena Than Those Of    Magnitude. Being Predicable

Of All Things Producing Indistinguishable Impressions, There Naturally

Grew Up Ideas Of    Equality In Weights, Sounds, Colours, Etc.; And Indeed

It Can Scarcely Be Doubted That The   Occasional Experience Of    Equal

Weights, Sounds, And Colours, Had A Share In Developing The   Abstract

Conception Of    Equality--That The   Ideas Of    Equality In Size, Relations,

Forces, Resistances, And Sensible Properties In General, Were Evolved

During The   Same Period. But However This May Be, It Is Clear That As

Fast As The   Notion Of    Equality Gained Definiteness, So Fast Did That

Lowest Kind Of    Quantitative Prevision Which Is Achieved Without Any

Instrumental Aid, Become Possible.

The Ability To Estimate, However Roughly, The   Amount Of    A Foreseen

Result, Implies The   Conception That It Will Be _Equal To_ A Certain

Imagined Quantity; And The   Correctness Of    The   Estimate Will Manifestly

Depend Upon The   Accuracy At Which The   Perceptions Of    Sensible Equality

Have Arrived. A Savage With A Piece Of    Stone In His Hand, And Another

Piece Lying Before Him Of    Greater Bulk Of    The   Same Kind (A Fact Which He

Infers From The   _Equality_ Of    The   Two In Colour And Texture) Knows About

What Effort He Must Put Forth To Raise This Other Piece; And He Judges

Accurately In Proportion To The   Accuracy With Which He Perceives That

The One Is Twice, Three Times, Four Times, Etc., As Large As The   Other;

That Is--In Proportion To The   Precision Of    His Ideas Of    Equality And

Number. And Here Let Us Not Omit To Notice That Even In These Vaguest Of

Quantitative Previsions, The   Conception Of    _Equality Of    Relations_ Is

Also Involved. For It Is Only In Virtue Of    An Undefined Perception That

The Relation Between Bulk And Weight In The   One Stone Is _Equal_ To The

Relation Between Bulk And Weight In The   Other, That Even The   Roughest

Approximation Can Be Made.

But How Came The   Transition From Those Uncertain Perceptions Of    Equality

Which The   Unaided Senses Give, To The   Certain Ones With Which Science

Deals? It Came By Placing The   Things Compared In Juxtaposition. Equality

Being Predicated Of    Things Which Give Us Indistinguishable Impressions,

And No Accurate Comparison Of    Impressions Being Possible Unless They

Occur In Immediate Succession, It Results That Exactness Of    Equality Is

Ascertainable In Proportion To The   Closeness Of    The   Compared Things.

Hence The   Fact That When We Wish To Judge Of    Two Shades Of    Colour

Whether They Are Alike Or Not, We Place Them Side By Side; Hence The

Fact That We Cannot, With Any Precision, Say Which Of    Two Allied Sounds

Is The   Louder, Or The   Higher In Pitch, Unless We Hear The   One

Immediately After The   Other; Hence The   Fact That To Estimate The   Ratio

Of Weights, We Take One In Each Hand, That We May Compare Their

Pressures By Rapidly Alternating In Thought From The   One To The   Other;

Hence The   Fact, That In A Piece Of    Music We Can Continue To Make Equal

Beats When The   First Beat Has Been Given, But Cannot Ensure Commencing

With The   Same Length Of    Beat On A Future Occasion; And Hence, Lastly,

The Fact, That Of    All Magnitudes, Those Of    _Linear Extension_ Are Those

Of Which The   Equality Is Most Accurately Ascertainable, And Those To

Which By Consequence All Others Have To Be Reduced. For It Is The

Peculiarity Of    Linear Extension That It Alone Allows Its Magnitudes To

Be Placed In _Absolute_ Juxtaposition, Or, Rather, In Coincident

Position; It Alone Can Test The   Equality Of    Two Magnitudes By Observing

Whether They Will Coalesce, As Two Equal Mathematical Lines Do, When

Placed Between The   Same Points; It Alone Can Test _Equality_ By Trying

Whether It Will Become _Identity_. Hence, Then, The   Fact, That All Exact

Science Is Reducible, By An Ultimate Analysis, To Results Measured In

Equal Units Of    Linear Extension.

Still It Remains To Be Noticed In What Manner This Determination Of

Equality By Comparison Of    Linear Magnitudes Originated. Once More May We

Perceive That Surrounding Natural Objects Supplied The   Needful Lessons.

From The   Beginning There Must Have Been A Constant Experience Of    Like

Things Placed Side By Side--Men Standing And Walking Together; Animals

From The   Same Herd; Fish From The   Same Shoal. And The   Ceaseless

Repetition Of    These Experiences Could Not Fail To Suggest The

Observation, That The   Nearer Together Any Objects Were, The   More Visible

Became Any Inequality Between Them. Hence The   Obvious Device Of    Putting

In Apposition Things Of    Which It Was Desired To Ascertain The   Relative

Magnitudes. Hence The   Idea Of    _Measure_. And Here We Suddenly Come Upon

A Group Of    Facts Which Afford A Solid Basis To The   Remainder Of    Our

Argument; While They Also Furnish Strong Evidence In Support Of    The

Foregoing Speculations. Those Who Look Sceptically On This Attempted

Rehabilitation Of    The   Earliest Epochs Of    Mental Development, And Who

More Especially Think That The   Derivation Of    So Many Primary Notions

From Organic Forms Is Somewhat Strained, Will Perhaps See More

Probability In The   Several Hypotheses That Have Been Ventured, On

Discovering That All Measures Of    _Extension_ And _Force_ Originated From

The Lengths And Weights Of    Organic Bodies; And All Measures Of    _Time_

From The   Periodic Phenomena Of    Either Organic Or Inorganic Bodies.

Thus, Among Linear Measures, The   Cubit Of    The   Hebrews Was The   _Length Of

The Forearm_ From The   Elbow To The   End Of    The   Middle Finger; And The

Smaller Scriptural Dimensions Are Expressed In _Hand-Breadths_ And

_Spans_. The   Egyptian Cubit, Which Was Similarly Derived, Was Divided

Into Digits, Which Were _Finger-Breadths_; And Each Finger-Breadth Was

More Definitely Expressed As Being Equal To Four _Grains Of    Barley_

Placed Breadthwise. Other Ancient Measures Were The   Orgyia Or _Stretch

Of The   Arms_, The   _Pace_, And The   _Palm_. So Persistent Has Been The   Use

Of These Natural Units Of    Length In The   East, That Even Now Some Of    The

Arabs Mete Out Cloth By The   Forearm. So, Too, Is It With European

Measures. The   _Foot_ Prevails As A Dimension Throughout Europe, And Has

Done Since The   Time Of    The   Romans, By Whom, Also, It Was Used: Its

Lengths In Different Places Varying Not Much More Than Men's Feet Vary.

The Heights Of    Horses Are Still Expressed In _Hands_. The   Inch Is The

Length Of    The   Terminal Joint Of    _The Thumb_; As Is Clearly Shown In

France, Where _Pouce_ Means Both Thumb And Inch. Then We Have The   Inch

Divided Into Three _Barley-Corns_.

So Completely, Indeed, Have These Organic Dimensions Served As The

Substrata Of    All Mensuration, That It Is Only By Means Of    Them That We

Can Form Any Estimate Of    Some Of    The   Ancient Distances. For Example, The

Length Of    A Degree On The   Earth's Surface, As Determined By The   Arabian

Astronomers Shortly After The   Death Of    Haroun-Al-Raschid, Was Fifty-Six

Of Their Miles. We Know Nothing Of    Their Mile Further Than That It Was

4000 Cubits; And Whether These Were Sacred Cubits Or Common Cubits,

Would Remain Doubtful, But That The   Length Of    The   Cubit Is Given As

Twenty-Seven Inches, And Each Inch Defined As The   Thickness Of    Six

Barley-Grains. Thus One Of    The   Earliest Measurements Of    A Degree Comes

Down To Us In Barley-Grains. Not Only Did Organic Lengths Furnish Those

Approximate Measures Which Satisfied Men's Needs In Ruder Ages, But