Javafx - Auto-scroll em TextArea

Veja abaixo uma imagem do programa em execução:

Veja abaixo o código do programa:

import javafx.application.Application;

import javafx.scene.Scene;

import javafx.scene.control.Button;

import javafx.scene.control.TextArea;

import javafx.scene.layout.BorderPane;

import javafx.stage.Stage;

public class PROJETO extends Application {

     @Override

     public void start ( Stage stage ) {

         try {

              stage.setTitle ( "TextArea javaFx - Auto-scroll" );

              BorderPane root = new BorderPane ( );

              Scene scene = new Scene ( root, 400, 400 );

              TextArea text = new TextArea ( );

              //Criando uma moldura no texarea

              root.setStyle ( "-fx-padding: 5;" + "-fx-border-style: solid inside;"

                        + "-fx-border-width: 18;" + "-fx-border-insets: 5;"

                        + "-fx-border-radius: 5;" + "-fx-border-color: blue;" );

              text.setStyle ( "-fx-font-family: Consolas;"

                        +"-fx-font-size: 20px;"

                        +"-fx-highlight-fill: #00ff00;"

                        +"-fx-highlight-text-fill: #000000;"

                        +"-fx-text-fill: black;"

                        +"-fx-background-color: #000000;");

              root.setCenter ( text );

              Button button = new Button ( "Clear" );

              button.setOnAction ( e -> {

                   // Limpa todo o texto

                   text.clear ( );

                   text.endOfNextWord ( );

              } );

              root.setBottom ( button );

              stage.setScene ( scene );

              stage.show ( );

         } catch ( Exception e ) {

              e.printStackTrace ( );

         }

     }

     public static void main ( String [ ] args ) {

         launch ( args );

     }

}